Results of Incomplete Gamma and Related Functions: Difference between revisions

From testwiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Tag: Replaced
 
(4 intermediate revisions by the same user not shown)
Line 1: Line 1:
{| class="wikitable sortable"
<div style="-moz-column-count:2; column-count:2;">
|-
; Notation : [[8.1|8.1 Special Notation]]<br>
! DLMF !! Formula !! Constraints !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
; Incomplete Gamma Functions : [[8.2|8.2 Definitions and Basic Properties]]<br>[[8.3|8.3 Graphics]]<br>[[8.4|8.4 Special Values]]<br>[[8.5|8.5 Confluent Hypergeometric Representations]]<br>[[8.6|8.6 Integral Representations]]<br>[[8.7|8.7 Series Expansions]]<br>[[8.8|8.8 Recurrence Relations and Derivatives]]<br>[[8.9|8.9 Continued Fractions]]<br>[[8.10|8.10 Inequalities]]<br>[[8.11|8.11 Asymptotic Approximations and Expansions]]<br>[[8.12|8.12 Uniform Asymptotic Expansions for Large Parameter]]<br>[[8.13|8.13 Zeros]]<br>[[8.14|8.14 Integrals]]<br>[[8.15|8.15 Sums]]<br>[[8.16|8.16 Generalizations]]<br>
|-
; Related Functions : [[8.17|8.17 Incomplete Beta Functions]]<br>[[8.18|8.18 Asymptotic Expansions of <math>\normincBetaI{x}@{a}{b}</math>]]<br>[[8.19|8.19 Generalized Exponential Integral]]<br>[[8.20|8.20 Asymptotic Expansions of <math>\genexpintE{p}@{z}</math>]]<br>[[8.21|8.21 Generalized Sine and Cosine Integrals]]<br>
| [https://dlmf.nist.gov/8.2.E1 8.2.E1] || [[Item:Q2480|<math>\incgamma@{a}{z} = \int_{0}^{z}t^{a-1}e^{-t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{z} = \int_{0}^{z}t^{a-1}e^{-t}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, z) = int((t)^(a - 1)* exp(- t), t = 0..z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, z] == Integrate[(t)^(a - 1)* Exp[- t], {t, 0, z}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 21] || Successful [Tested: 21]
; Applications : [[8.22|8.22 Mathematical Applications]]<br>[[8.23|8.23 Statistical Applications]]<br>[[8.24|8.24 Physical Applications]]<br>
|-
; Computation : [[8.25|8.25 Methods of Computation]]<br>[[8.26|8.26 Tables]]<br>[[8.27|8.27 Approximations]]<br>[[8.28|8.28 Software]]<br>
| [https://dlmf.nist.gov/8.2.E2 8.2.E2] || [[Item:Q2481|<math>\incGamma@{a}{z} = \int_{z}^{\infty}t^{a-1}e^{-t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{z} = \int_{z}^{\infty}t^{a-1}e^{-t}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(a, z) = int((t)^(a - 1)* exp(- t), t = z..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, z] == Integrate[(t)^(a - 1)* Exp[- t], {t, z, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [14 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
</div>
Test Values: {a = -2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {a = -2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 42]
|-
| [https://dlmf.nist.gov/8.2.E3 8.2.E3] || [[Item:Q2482|<math>\incgamma@{a}{z}+\incGamma@{a}{z} = \EulerGamma@{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{z}+\incGamma@{a}{z} = \EulerGamma@{a}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, z)+ GAMMA(a, z) = GAMMA(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, z]+ Gamma[a, z] == Gamma[a]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/8.2#Ex1 8.2#Ex1] || [[Item:Q2483|<math>\normincGammaP@{a}{z} = \frac{\incgamma@{a}{z}}{\EulerGamma@{a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincGammaP@{a}{z} = \frac{\incgamma@{a}{z}}{\EulerGamma@{a}}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = (GAMMA(a)-GAMMA(a, z))/(GAMMA(a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>GammaRegularized[a, 0, z] == Divide[Gamma[a, 0, z],Gamma[a]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.2#Ex2 8.2#Ex2] || [[Item:Q2484|<math>\normincGammaQ@{a}{z} = \frac{\incGamma@{a}{z}}{\EulerGamma@{a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincGammaQ@{a}{z} = \frac{\incGamma@{a}{z}}{\EulerGamma@{a}}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, z)/GAMMA(a) = (GAMMA(a, z))/(GAMMA(a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>GammaRegularized[a, z] == Divide[Gamma[a, z],Gamma[a]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.2.E5 8.2.E5] || [[Item:Q2485|<math>\normincGammaP@{a}{z}+\normincGammaQ@{a}{z} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincGammaP@{a}{z}+\normincGammaQ@{a}{z} = 1</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>(GAMMA(a)-GAMMA(a, z))/GAMMA(a)+ GAMMA(a, z)/GAMMA(a) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>GammaRegularized[a, 0, z]+ GammaRegularized[a, z] == 1</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.2.E6 8.2.E6] || [[Item:Q2486|<math>\scincgamma@{a}{z} = z^{-a}\normincGammaP@{a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\scincgamma@{a}{z} = z^{-a}\normincGammaP@{a}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = (z)^(- a)* (GAMMA(a)-GAMMA(a, z))/GAMMA(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Successful || Missing Macro Error || Skip - symbolical successful subtest || -
|-
| [https://dlmf.nist.gov/8.2.E6 8.2.E6] || [[Item:Q2486|<math>z^{-a}\normincGammaP@{a}{z} = \frac{z^{-a}}{\EulerGamma@{a}}\incgamma@{a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z^{-a}\normincGammaP@{a}{z} = \frac{z^{-a}}{\EulerGamma@{a}}\incgamma@{a}{z}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>(z)^(- a)* (GAMMA(a)-GAMMA(a, z))/GAMMA(a) = ((z)^(- a))/(GAMMA(a))*GAMMA(a)-GAMMA(a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(z)^(- a)* GammaRegularized[a, 0, z] == Divide[(z)^(- a),Gamma[a]]*Gamma[a, 0, z]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .2669922311+.3869142026*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.493684189-.9843316111*I
Test Values: {a = 1.5, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.2.E7 8.2.E7] || [[Item:Q2487|<math>\scincgamma@{a}{z} = \frac{1}{\EulerGamma@{a}}\int_{0}^{1}t^{a-1}e^{-zt}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\scincgamma@{a}{z} = \frac{1}{\EulerGamma@{a}}\int_{0}^{1}t^{a-1}e^{-zt}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = (1)/(GAMMA(a))*int((t)^(a - 1)* exp(- z*t), t = 0..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Failure || Missing Macro Error || Successful [Tested: 21] || -
|-
| [https://dlmf.nist.gov/8.2.E8 8.2.E8] || [[Item:Q2488|<math>\incgamma@{a}{ze^{2\pi mi}} = e^{2\pi mia}\incgamma@{a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{ze^{2\pi mi}} = e^{2\pi mia}\incgamma@{a}{z}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, z*exp(2*Pi*m*I)) = exp(2*Pi*m*I*a)*GAMMA(a)-GAMMA(a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, z*Exp[2*Pi*m*I]] == Exp[2*Pi*m*I*a]*Gamma[a, 0, z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [28 / 63]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.772453851+.14e-8*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, m = 1, a = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.772453851+.62e-8*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, m = 3, a = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.2.E9 8.2.E9] || [[Item:Q2489|<math>\incGamma@{a}{ze^{2\pi mi}} = e^{2\pi mia}\incGamma@{a}{z}+(1-e^{2\pi mia})\EulerGamma@{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{ze^{2\pi mi}} = e^{2\pi mia}\incGamma@{a}{z}+(1-e^{2\pi mia})\EulerGamma@{a}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, z*exp(2*Pi*m*I)) = exp(2*Pi*m*I*a)*GAMMA(a, z)+(1 - exp(2*Pi*m*I*a))*GAMMA(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, z*Exp[2*Pi*m*I]] == Exp[2*Pi*m*I*a]*Gamma[a, z]+(1 - Exp[2*Pi*m*I*a])*Gamma[a]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [28 / 63]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.701776495-.3980325655*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, m = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.701776493-.3980325669*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, m = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [28 / 63]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.7017764946044596, -0.3980325648566406]
Test Values: {Rule[a, 1.5], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.7017764946044598, -0.39803256485664035]
Test Values: {Rule[a, 1.5], Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.2.E10 8.2.E10] || [[Item:Q2490|<math>e^{-\pi ia}\incGamma@{a}{ze^{\pi i}}-e^{\pi ia}\incGamma@{a}{ze^{-\pi i}} = -\frac{2\pi i}{\EulerGamma@{1-a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-\pi ia}\incGamma@{a}{ze^{\pi i}}-e^{\pi ia}\incGamma@{a}{ze^{-\pi i}} = -\frac{2\pi i}{\EulerGamma@{1-a}}</syntaxhighlight> || <math>\realpart@@{1-a} > 0</math> || <syntaxhighlight lang=mathematica>exp(- Pi*I*a)*GAMMA(a, z*exp(Pi*I))- exp(Pi*I*a)*GAMMA(a, z*exp(- Pi*I)) = -(2*Pi*I)/(GAMMA(1 - a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[- Pi*I*a]*Gamma[a, z*Exp[Pi*I]]- Exp[Pi*I*a]*Gamma[a, z*Exp[- Pi*I]] == -Divide[2*Pi*I,Gamma[1 - a]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [28 / 28]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -2.710645106+2.690107924*I
Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .1295806364+5.171352915*I
Test Values: {a = -1.5, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [28 / 28]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-2.710645107283499, 2.690107923521634]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.12958063668571548, 5.171352913200156]
Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.2.E11 8.2.E11] || [[Item:Q2491|<math>\incGamma@{a}{ze^{+\pi i}} = \EulerGamma@{a}(1-z^{a}e^{+\pi ia}\scincgamma@{a}{-z})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{ze^{+\pi i}} = \EulerGamma@{a}(1-z^{a}e^{+\pi ia}\scincgamma@{a}{-z})</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, z*exp(+ Pi*I)) = GAMMA(a)*(1 - (z)^(a)* exp(+ Pi*I*a)*(- z)^(-(a))*(GAMMA(a)-GAMMA(a, - z))/GAMMA(a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Failure || Missing Macro Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.046852240-1.006153525*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.4563214597+.8560373719*I
Test Values: {a = 1.5, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || -
|-
| [https://dlmf.nist.gov/8.2.E11 8.2.E11] || [[Item:Q2491|<math>\incGamma@{a}{ze^{-\pi i}} = \EulerGamma@{a}(1-z^{a}e^{-\pi ia}\scincgamma@{a}{-z})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{ze^{-\pi i}} = \EulerGamma@{a}(1-z^{a}e^{-\pi ia}\scincgamma@{a}{-z})</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, z*exp(- Pi*I)) = GAMMA(a)*(1 - (z)^(a)* exp(- Pi*I*a)*(- z)^(-(a))*(GAMMA(a)-GAMMA(a, - z))/GAMMA(a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Failure || Missing Macro Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.510498964-.9249588863*I
Test Values: {a = 1.5, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.7017764949-.3980325647*I
Test Values: {a = 1.5, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || -
|-
| [https://dlmf.nist.gov/8.2.E12 8.2.E12] || [[Item:Q2492|<math>\deriv[2]{w}{z}+\left(1+\frac{1-a}{z}\right)\deriv{w}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{z}+\left(1+\frac{1-a}{z}\right)\deriv{w}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [z$(2)])+(1 +(1 - a)/(z))*diff(w, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 2}]+(1 +Divide[1 - a,z])*D[w, z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 300]
|-
| [https://dlmf.nist.gov/8.2.E13 8.2.E13] || [[Item:Q2493|<math>\deriv[2]{w}{z}-\left(1+\frac{1-a}{z}\right)\deriv{w}{z}+\frac{1-a}{z^{2}}w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{z}-\left(1+\frac{1-a}{z}\right)\deriv{w}{z}+\frac{1-a}{z^{2}}w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [z$(2)])-(1 +(1 - a)/(z))*diff(w, z)+(1 - a)/((z)^(2))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 2}]-(1 +Divide[1 - a,z])*D[w, z]+Divide[1 - a,(z)^(2)]*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.165063509-1.250000000*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -2.165063509+1.250000000*I
Test Values: {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[2.1650635094610964, -1.25]
Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-2.1650635094610964, 1.2500000000000004]
Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.4.E1 8.4.E1] || [[Item:Q2495|<math>\incgamma@{\tfrac{1}{2}}{z^{2}} = 2\int_{0}^{z}e^{-t^{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{\tfrac{1}{2}}{z^{2}} = 2\int_{0}^{z}e^{-t^{2}}\diff{t}</syntaxhighlight> || <math>\realpart@@{\tfrac{1}{2}} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA((1)/(2))-GAMMA((1)/(2), (z)^(2)) = 2*int(exp(- (t)^(2)), t = 0..z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[Divide[1,2], 0, (z)^(2)] == 2*Integrate[Exp[- (t)^(2)], {t, 0, z}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.465949776-3.038201708*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.197911286+.8974462698*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[3.4659497742269214, -3.038201707267986]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[3.197911285535813, 0.8974462701863266]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div>
|-
| [https://dlmf.nist.gov/8.4.E1 8.4.E1] || [[Item:Q2495|<math>2\int_{0}^{z}e^{-t^{2}}\diff{t} = \sqrt{\pi}\erf@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2\int_{0}^{z}e^{-t^{2}}\diff{t} = \sqrt{\pi}\erf@{z}</syntaxhighlight> || <math>\realpart@@{\tfrac{1}{2}} > 0</math> || <syntaxhighlight lang=mathematica>2*int(exp(- (t)^(2)), t = 0..z) = sqrt(Pi)*erf(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>2*Integrate[Exp[- (t)^(2)], {t, 0, z}, GenerateConditions->None] == Sqrt[Pi]*Erf[z]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/8.4.E2 8.4.E2] || [[Item:Q2496|<math>\scincgamma@{a}{0} = \frac{1}{\EulerGamma@{a+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\scincgamma@{a}{0} = \frac{1}{\EulerGamma@{a+1}}</syntaxhighlight> || <math>\realpart@@{a+1} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>(0)^(-(a))*(GAMMA(a)-GAMMA(a, 0))/GAMMA(a) = (1)/(GAMMA(a + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Failure || Missing Macro Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.7522527782
Test Values: {a = 1.5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.128379167
Test Values: {a = .5}</syntaxhighlight><br>... skip entries to safe data</div></div> || -
|-
| [https://dlmf.nist.gov/8.4.E3 8.4.E3] || [[Item:Q2497|<math>\scincgamma@{\tfrac{1}{2}}{-z^{2}} = \frac{2e^{z^{2}}}{z\sqrt{\pi}}\DawsonsintF@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\scincgamma@{\tfrac{1}{2}}{-z^{2}} = \frac{2e^{z^{2}}}{z\sqrt{\pi}}\DawsonsintF@{z}</syntaxhighlight> || <math>\realpart@@{\tfrac{1}{2}} > 0</math> || <syntaxhighlight lang=mathematica>(- (z)^(2))^(-((1)/(2)))*(GAMMA((1)/(2))-GAMMA((1)/(2), - (z)^(2)))/GAMMA((1)/(2)) = (2*exp((z)^(2)))/(z*sqrt(Pi))*dawson(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Successful || Missing Macro Error || - || -
|-
| [https://dlmf.nist.gov/8.4.E4 8.4.E4] || [[Item:Q2498|<math>\incGamma@{0}{z} = \int_{z}^{\infty}t^{-1}e^{-t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{0}{z} = \int_{z}^{\infty}t^{-1}e^{-t}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(0, z) = int((t)^(- 1)* exp(- t), t = z..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[0, z] == Integrate[(t)^(- 1)* Exp[- t], {t, z, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/8.4.E4 8.4.E4] || [[Item:Q2498|<math>\int_{z}^{\infty}t^{-1}e^{-t}\diff{t} = \expintE@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{z}^{\infty}t^{-1}e^{-t}\diff{t} = \expintE@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((t)^(- 1)* exp(- t), t = z..infinity) = Ei(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^(- 1)* Exp[- t], {t, z, Infinity}, GenerateConditions->None] == ExpIntegralE[1, z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.393548628-1.498247032*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.8944744989-3.773814377*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/8.4.E5 8.4.E5] || [[Item:Q2499|<math>\incGamma@{1}{z} = e^{-z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{1}{z} = e^{-z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(1, z) = exp(- z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[1, z] == Exp[- z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/8.4.E6 8.4.E6] || [[Item:Q2500|<math>\incGamma@{\tfrac{1}{2}}{z^{2}} = 2\int_{z}^{\infty}e^{-t^{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{\tfrac{1}{2}}{z^{2}} = 2\int_{z}^{\infty}e^{-t^{2}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA((1)/(2), (z)^(2)) = 2*int(exp(- (t)^(2)), t = z..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[Divide[1,2], (z)^(2)] == 2*Integrate[Exp[- (t)^(2)], {t, z, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -3.465949776+3.038201708*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -3.197911286-.8974462698*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-3.4659497742269214, 3.038201707267986]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-3.1979112855358127, -0.8974462701863266]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div>
|-
| [https://dlmf.nist.gov/8.4.E6 8.4.E6] || [[Item:Q2500|<math>2\int_{z}^{\infty}e^{-t^{2}}\diff{t} = \sqrt{\pi}\erfc@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2\int_{z}^{\infty}e^{-t^{2}}\diff{t} = \sqrt{\pi}\erfc@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>2*int(exp(- (t)^(2)), t = z..infinity) = sqrt(Pi)*erfc(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>2*Integrate[Exp[- (t)^(2)], {t, z, Infinity}, GenerateConditions->None] == Sqrt[Pi]*Erfc[z]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/8.4.E7 8.4.E7] || [[Item:Q2501|<math>\incgamma@{n+1}{z} = n!(1-e^{-z}e_{n}(z))</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{n+1}{z} = n!(1-e^{-z}e_{n}(z))</syntaxhighlight> || <math>\realpart@@{n+1} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(n + 1)-GAMMA(n + 1, z) = factorial(n)*(1 - exp(- z)*exp(1)[n]*(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[n + 1, 0, z] == (n)!*(1 - Exp[- z]*Subscript[E, n]*(z))</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.7896317094254578, 0.19173078621885742], Times[Complex[0.42050290937849244, 0.009925196319850484], Subscript[2.718281828459045, 1]]]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.06153297742196945, 0.16461464559793018], Times[-2.0, Plus[1.0, Times[Complex[-0.42050290937849244, -0.009925196319850484], Subscript[2.718281828459045, 2]]]]]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.4.E8 8.4.E8] || [[Item:Q2502|<math>\incGamma@{n+1}{z} = n!e^{-z}e_{n}(z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{n+1}{z} = n!e^{-z}e_{n}(z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(n + 1, z) = factorial(n)*exp(- z)*exp(1)[n]*(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[n + 1, z] == (n)!*Exp[- z]*Subscript[E, n]*(z)</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.7896317094254578, -0.19173078621885742], Times[Complex[-0.42050290937849244, -0.009925196319850484], Subscript[2.718281828459045, 1]]]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.9384670225780305, -0.16461464559793018], Times[Complex[-0.8410058187569849, -0.019850392639700967], Subscript[2.718281828459045, 2]]]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.4.E9 8.4.E9] || [[Item:Q2503|<math>\normincGammaP@{n+1}{z} = 1-e^{-z}e_{n}(z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincGammaP@{n+1}{z} = 1-e^{-z}e_{n}(z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(GAMMA(n + 1)-GAMMA(n + 1, z))/GAMMA(n + 1) = 1 - exp(- z)*exp(1)[n]*(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>GammaRegularized[n + 1, 0, z] == 1 - Exp[- z]*Subscript[E, n]*(z)</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.7896317094254579, 0.1917307862188573], Times[Complex[0.42050290937849244, 0.009925196319850484], Subscript[2.718281828459045, 1]]]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.9692335112890152, 0.08230732279896512], Times[Complex[0.42050290937849244, 0.009925196319850484], Subscript[2.718281828459045, 2]]]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.4.E10 8.4.E10] || [[Item:Q2504|<math>\normincGammaQ@{n+1}{z} = e^{-z}e_{n}(z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincGammaQ@{n+1}{z} = e^{-z}e_{n}(z)</syntaxhighlight> || <math>\realpart@@{n+1} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(n + 1, z)/GAMMA(n + 1) = exp(- z)*exp(1)[n]*(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>GammaRegularized[n + 1, z] == Exp[- z]*Subscript[E, n]*(z)</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.7896317094254579, -0.1917307862188573], Times[Complex[-0.42050290937849244, -0.009925196319850484], Subscript[2.718281828459045, 1]]]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.9692335112890152, -0.08230732279896512], Times[Complex[-0.42050290937849244, -0.009925196319850484], Subscript[2.718281828459045, 2]]]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.4.E12 8.4.E12] || [[Item:Q2506|<math>\scincgamma@{-n}{z} = z^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\scincgamma@{-n}{z} = z^{n}</syntaxhighlight> || <math>\realpart@@{-n} > 0</math> || <syntaxhighlight lang=mathematica>(z)^(-(- n))*(GAMMA(- n)-GAMMA(- n, z))/GAMMA(- n) = (z)^(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Failure || Missing Macro Error || Error || -
|-
| [https://dlmf.nist.gov/8.4.E13 8.4.E13] || [[Item:Q2507|<math>\incGamma@{1-n}{z} = z^{1-n}\genexpintE{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{1-n}{z} = z^{1-n}\genexpintE{n}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(1 - n, z) = (z)^(1 - n)* Ei(n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[1 - n, z] == (z)^(1 - n)* ExpIntegralE[n, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.4.E14 8.4.E14] || [[Item:Q2508|<math>\normincGammaQ@{n+\tfrac{1}{2}}{z^{2}} = \erfc@{z}+\frac{e^{-z^{2}}}{\sqrt{\pi}}\sum_{k=1}^{n}\frac{z^{2k-1}}{\Pochhammersym{\tfrac{1}{2}}{k}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincGammaQ@{n+\tfrac{1}{2}}{z^{2}} = \erfc@{z}+\frac{e^{-z^{2}}}{\sqrt{\pi}}\sum_{k=1}^{n}\frac{z^{2k-1}}{\Pochhammersym{\tfrac{1}{2}}{k}}</syntaxhighlight> || <math>\realpart@@{n+\tfrac{1}{2}} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(n +(1)/(2), (z)^(2))/GAMMA(n +(1)/(2)) = erfc(z)+(exp(- (z)^(2)))/(sqrt(Pi))*sum(((z)^(2*k - 1))/(pochhammer((1)/(2), k)), k = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>GammaRegularized[n +Divide[1,2], (z)^(2)] == Erfc[z]+Divide[Exp[- (z)^(2)],Sqrt[Pi]]*Sum[Divide[(z)^(2*k - 1),Pochhammer[Divide[1,2], k]], {k, 1, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.704415567+1.043704337*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .97393781e-1-.8458491548*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.7044155650581054, 1.0437043365740406]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.09739377924871273, -0.8458491528064774]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.4.E15 8.4.E15] || [[Item:Q2509|<math>\incGamma@{-n}{z} = \frac{(-1)^{n}}{n!}\left(\expintE@{z}-e^{-z}\sum_{k=0}^{n-1}\frac{(-1)^{k}k!}{z^{k+1}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{-n}{z} = \frac{(-1)^{n}}{n!}\left(\expintE@{z}-e^{-z}\sum_{k=0}^{n-1}\frac{(-1)^{k}k!}{z^{k+1}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(- n, z) = ((- 1)^(n))/(factorial(n))*(Ei(z)- exp(- z)*sum(((- 1)^(k)* factorial(k))/((z)^(k + 1)), k = 0..n - 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[- n, z] == Divide[(- 1)^(n),(n)!]*(ExpIntegralE[1, z]- Exp[- z]*Sum[Divide[(- 1)^(k)* (k)!,(z)^(k + 1)], {k, 0, n - 1}, GenerateConditions->None])</syntaxhighlight> || Failure || Failure || Manual Skip! || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.5.E1 8.5.E1] || [[Item:Q2510|<math>\incgamma@{a}{z} = a^{-1}z^{a}e^{-z}\KummerconfhyperM@{1}{1+a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{z} = a^{-1}z^{a}e^{-z}\KummerconfhyperM@{1}{1+a}{z}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, z) = (a)^(- 1)* (z)^(a)* exp(- z)*KummerM(1, 1 + a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, z] == (a)^(- 1)* (z)^(a)* Exp[- z]*Hypergeometric1F1[1, 1 + a, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/8.5.E1 8.5.E1] || [[Item:Q2510|<math>a^{-1}z^{a}e^{-z}\KummerconfhyperM@{1}{1+a}{z} = a^{-1}z^{a}\KummerconfhyperM@{a}{1+a}{-z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>a^{-1}z^{a}e^{-z}\KummerconfhyperM@{1}{1+a}{z} = a^{-1}z^{a}\KummerconfhyperM@{a}{1+a}{-z}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>(a)^(- 1)* (z)^(a)* exp(- z)*KummerM(1, 1 + a, z) = (a)^(- 1)* (z)^(a)* KummerM(a, 1 + a, - z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(a)^(- 1)* (z)^(a)* Exp[- z]*Hypergeometric1F1[1, 1 + a, z] == (a)^(- 1)* (z)^(a)* Hypergeometric1F1[a, 1 + a, - z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/8.5.E2 8.5.E2] || [[Item:Q2511|<math>\scincgamma@{a}{z} = e^{-z}\OlverconfhyperM@{1}{1+a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\scincgamma@{a}{z} = e^{-z}\OlverconfhyperM@{1}{1+a}{z}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = exp(- z)*KummerM(1, 1 + a, z)/GAMMA(1 + a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Successful || Missing Macro Error || - || -
|-
| [https://dlmf.nist.gov/8.5.E2 8.5.E2] || [[Item:Q2511|<math>e^{-z}\OlverconfhyperM@{1}{1+a}{z} = \OlverconfhyperM@{a}{1+a}{-z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-z}\OlverconfhyperM@{1}{1+a}{z} = \OlverconfhyperM@{a}{1+a}{-z}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>exp(- z)*KummerM(1, 1 + a, z)/GAMMA(1 + a) = KummerM(a, 1 + a, - z)/GAMMA(1 + a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[- z]*Hypergeometric1F1Regularized[1, 1 + a, z] == Hypergeometric1F1Regularized[a, 1 + a, - z]</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.5.E3 8.5.E3] || [[Item:Q2512|<math>\incGamma@{a}{z} = e^{-z}\KummerconfhyperU@{1-a}{1-a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{z} = e^{-z}\KummerconfhyperU@{1-a}{1-a}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(a, z) = exp(- z)*KummerU(1 - a, 1 - a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, z] == Exp[- z]*HypergeometricU[1 - a, 1 - a, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
|-
| [https://dlmf.nist.gov/8.5.E3 8.5.E3] || [[Item:Q2512|<math>e^{-z}\KummerconfhyperU@{1-a}{1-a}{z} = z^{a}e^{-z}\KummerconfhyperU@{1}{1+a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-z}\KummerconfhyperU@{1-a}{1-a}{z} = z^{a}e^{-z}\KummerconfhyperU@{1}{1+a}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>exp(- z)*KummerU(1 - a, 1 - a, z) = (z)^(a)* exp(- z)*KummerU(1, 1 + a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[- z]*HypergeometricU[1 - a, 1 - a, z] == (z)^(a)* Exp[- z]*HypergeometricU[1, 1 + a, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
|-
| [https://dlmf.nist.gov/8.5.E4 8.5.E4] || [[Item:Q2513|<math>\incgamma@{a}{z} = a^{-1}z^{\frac{1}{2}a-\frac{1}{2}}e^{-\frac{1}{2}z}\WhittakerconfhyperM{\frac{1}{2}a-\frac{1}{2}}{\frac{1}{2}a}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{z} = a^{-1}z^{\frac{1}{2}a-\frac{1}{2}}e^{-\frac{1}{2}z}\WhittakerconfhyperM{\frac{1}{2}a-\frac{1}{2}}{\frac{1}{2}a}@{z}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, z) = (a)^(- 1)* (z)^((1)/(2)*a -(1)/(2))* exp(-(1)/(2)*z)*WhittakerM((1)/(2)*a -(1)/(2), (1)/(2)*a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, z] == (a)^(- 1)* (z)^(Divide[1,2]*a -Divide[1,2])* Exp[-Divide[1,2]*z]*WhittakerM[Divide[1,2]*a -Divide[1,2], Divide[1,2]*a, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.5.E5 8.5.E5] || [[Item:Q2514|<math>\incGamma@{a}{z} = e^{-\frac{1}{2}z}z^{\frac{1}{2}a-\frac{1}{2}}\WhittakerconfhyperW{\frac{1}{2}a-\frac{1}{2}}{\frac{1}{2}a}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{z} = e^{-\frac{1}{2}z}z^{\frac{1}{2}a-\frac{1}{2}}\WhittakerconfhyperW{\frac{1}{2}a-\frac{1}{2}}{\frac{1}{2}a}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(a, z) = exp(-(1)/(2)*z)*(z)^((1)/(2)*a -(1)/(2))* WhittakerW((1)/(2)*a -(1)/(2), (1)/(2)*a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, z] == Exp[-Divide[1,2]*z]*(z)^(Divide[1,2]*a -Divide[1,2])* WhittakerW[Divide[1,2]*a -Divide[1,2], Divide[1,2]*a, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
|-
| [https://dlmf.nist.gov/8.6.E1 8.6.E1] || [[Item:Q2515|<math>\incgamma@{a}{z} = \frac{z^{a}}{\sin@{\pi a}}\int_{0}^{\pi}e^{z\cos@@{t}}\cos@{at+z\sin@@{t}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{z} = \frac{z^{a}}{\sin@{\pi a}}\int_{0}^{\pi}e^{z\cos@@{t}}\cos@{at+z\sin@@{t}}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, z) = ((z)^(a))/(sin(Pi*a))*int(exp(z*cos(t))*cos(a*t + z*sin(t)), t = 0..Pi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, z] == Divide[(z)^(a),Sin[Pi*a]]*Integrate[Exp[z*Cos[t]]*Cos[a*t + z*Sin[t]], {t, 0, Pi}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [14 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.922649672+.1964472815*I
Test Values: {a = .5, z = 1/2*3^(1/2)+1/2*I, a = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.511118576+1.941926371*I
Test Values: {a = .5, z = -1/2+1/2*I*3^(1/2), a = 3/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.6.E2 8.6.E2] || [[Item:Q2516|<math>\incgamma@{a}{z} = z^{\frac{1}{2}a}\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}a-1}\BesselJ{a}@{2\sqrt{zt}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{z} = z^{\frac{1}{2}a}\int_{0}^{\infty}e^{-t}t^{\frac{1}{2}a-1}\BesselJ{a}@{2\sqrt{zt}}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, z) = (z)^((1)/(2)*a)* int(exp(- t)*(t)^((1)/(2)*a - 1)* BesselJ(a, 2*sqrt(z*t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, z] == (z)^(Divide[1,2]*a)* Integrate[Exp[- t]*(t)^(Divide[1,2]*a - 1)* BesselJ[a, 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 21] || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.6.E3 8.6.E3] || [[Item:Q2517|<math>\incgamma@{a}{z} = z^{a}\int_{0}^{\infty}\exp@{-at-ze^{-t}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{z} = z^{a}\int_{0}^{\infty}\exp@{-at-ze^{-t}}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, z) = (z)^(a)* int(exp(- a*t - z*exp(- t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, z] == (z)^(a)* Integrate[Exp[- a*t - z*Exp[- t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 21] || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.6.E4 8.6.E4] || [[Item:Q2518|<math>\incGamma@{a}{z} = \frac{z^{a}e^{-z}}{\EulerGamma@{1-a}}\int_{0}^{\infty}\frac{t^{-a}e^{-t}}{z+t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{z} = \frac{z^{a}e^{-z}}{\EulerGamma@{1-a}}\int_{0}^{\infty}\frac{t^{-a}e^{-t}}{z+t}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \pi, \realpart@@{a} < 1, \realpart@@{1-a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, z) = ((z)^(a)* exp(- z))/(GAMMA(1 - a))*int(((t)^(- a)* exp(- t))/(z + t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, z] == Divide[(z)^(a)* Exp[- z],Gamma[1 - a]]*Integrate[Divide[(t)^(- a)* Exp[- t],z + t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 28]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {a = -1.5, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 28]
|-
| [https://dlmf.nist.gov/8.6.E5 8.6.E5] || [[Item:Q2519|<math>\incGamma@{a}{z} = z^{a}e^{-z}\int_{0}^{\infty}\frac{e^{-zt}}{(1+t)^{1-a}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{z} = z^{a}e^{-z}\int_{0}^{\infty}\frac{e^{-zt}}{(1+t)^{1-a}}\diff{t}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, z) = (z)^(a)* exp(- z)*int((exp(- z*t))/((1 + t)^(1 - a)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, z] == (z)^(a)* Exp[- z]*Integrate[Divide[Exp[- z*t],(1 + t)^(1 - a)], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 30]
|-
| [https://dlmf.nist.gov/8.6.E6 8.6.E6] || [[Item:Q2520|<math>\incGamma@{a}{z} = \frac{2z^{\frac{1}{2}a}e^{-z}}{\EulerGamma@{1-a}}\int_{0}^{\infty}e^{-t}t^{-\frac{1}{2}a}\modBesselK{a}@{2\sqrt{zt}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{z} = \frac{2z^{\frac{1}{2}a}e^{-z}}{\EulerGamma@{1-a}}\int_{0}^{\infty}e^{-t}t^{-\frac{1}{2}a}\modBesselK{a}@{2\sqrt{zt}}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} < 1, \realpart@@{1-a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, z) = (2*(z)^((1)/(2)*a)* exp(- z))/(GAMMA(1 - a))*int(exp(- t)*(t)^(-(1)/(2)*a)* BesselK(a, 2*sqrt(z*t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, z] == Divide[2*(z)^(Divide[1,2]*a)* Exp[- z],Gamma[1 - a]]*Integrate[Exp[- t]*(t)^(-Divide[1,2]*a)* BesselK[a, 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 28]
|-
| [https://dlmf.nist.gov/8.6.E7 8.6.E7] || [[Item:Q2521|<math>\incGamma@{a}{z} = z^{a}\int_{0}^{\infty}\exp@{at-ze^{t}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{z} = z^{a}\int_{0}^{\infty}\exp@{at-ze^{t}}\diff{t}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, z) = (z)^(a)* int(exp(a*t - z*exp(t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, z] == (z)^(a)* Integrate[Exp[a*t - z*Exp[t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 30] || Successful [Tested: 30]
|-
| [https://dlmf.nist.gov/8.6.E8 8.6.E8] || [[Item:Q2522|<math>\incgamma@{a}{z} = \frac{-\iunit z^{a}}{2\sin@{\pi a}}\int_{-1}^{(0+)}t^{a-1}e^{zt}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{z} = \frac{-\iunit z^{a}}{2\sin@{\pi a}}\int_{-1}^{(0+)}t^{a-1}e^{zt}\diff{t}</syntaxhighlight> || <math>z \neq 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, z) = (- I*(z)^(a))/(2*sin(Pi*a))*int((t)^(a - 1)* exp(z*t), t = - 1..(0 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, z] == Divide[- I*(z)^(a),2*Sin[Pi*a]]*Integrate[(t)^(a - 1)* Exp[z*t], {t, - 1, (0 +)}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
|-
| [https://dlmf.nist.gov/8.6.E9 8.6.E9] || [[Item:Q2523|<math>\incGamma@{-a}{ze^{+\pi i}} = \frac{e^{z}e^{-\pi\iunit a}}{\EulerGamma@{1+a}}\int_{0}^{\infty}\frac{t^{a}e^{-zt}}{t-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{-a}{ze^{+\pi i}} = \frac{e^{z}e^{-\pi\iunit a}}{\EulerGamma@{1+a}}\int_{0}^{\infty}\frac{t^{a}e^{-zt}}{t-1}\diff{t}</syntaxhighlight> || <math>\realpart@@{z} > 0, \realpart@@{a} > -1, \realpart@@{1+a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(- a, z*exp(+ Pi*I)) = (exp(z)*exp(- Pi*I*a))/(GAMMA(1 + a))*int(((t)^(a)* exp(- z*t))/(t - 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[- a, z*Exp[+ Pi*I]] == Divide[Exp[z]*Exp[- Pi*I*a],Gamma[1 + a]]*Integrate[Divide[(t)^(a)* Exp[- z*t],t - 1], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 20]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {a = 1.5, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.6.E9 8.6.E9] || [[Item:Q2523|<math>\incGamma@{-a}{ze^{-\pi i}} = \frac{e^{z}e^{+\pi\iunit a}}{\EulerGamma@{1+a}}\int_{0}^{\infty}\frac{t^{a}e^{-zt}}{t-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{-a}{ze^{-\pi i}} = \frac{e^{z}e^{+\pi\iunit a}}{\EulerGamma@{1+a}}\int_{0}^{\infty}\frac{t^{a}e^{-zt}}{t-1}\diff{t}</syntaxhighlight> || <math>\realpart@@{z} > 0, \realpart@@{a} > -1, \realpart@@{1+a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(- a, z*exp(- Pi*I)) = (exp(z)*exp(+ Pi*I*a))/(GAMMA(1 + a))*int(((t)^(a)* exp(- z*t))/(t - 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[- a, z*Exp[- Pi*I]] == Divide[Exp[z]*Exp[+ Pi*I*a],Gamma[1 + a]]*Integrate[Divide[(t)^(a)* Exp[- z*t],t - 1], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 20]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {a = 1.5, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.6.E10 8.6.E10] || [[Item:Q2524|<math>\incgamma@{a}{z} = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\EulerGamma@{s}}{a-s}z^{a-s}\diff{s}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{z} = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\EulerGamma@{s}}{a-s}z^{a-s}\diff{s}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{2}\pi, \realpart@@{s} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, z) = (1)/(2*Pi*I)*int((GAMMA(s))/(a - s)*(z)^(a - s), s = c - I*infinity..c + I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, z] == Divide[1,2*Pi*I]*Integrate[Divide[Gamma[s],a - s]*(z)^(a - s), {s, c - I*Infinity, c + I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [90 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .3508882474+.1990162824*I
Test Values: {a = 1.5, c = -1.5, z = 1/2*3^(1/2)+1/2*I, a = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2281607298-.4280186861*I
Test Values: {a = 1.5, c = -1.5, z = 1/2-1/2*I*3^(1/2), a = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.6.E11 8.6.E11] || [[Item:Q2525|<math>\incGamma@{a}{z} = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\EulerGamma@{s+a}\frac{z^{-s}}{s}\diff{s}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{z} = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\EulerGamma@{s+a}\frac{z^{-s}}{s}\diff{s}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{2}\pi, \realpart@@{s+a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, z) = (1)/(2*Pi*I)*int(GAMMA(s + a)*((z)^(- s))/(s), s = c - I*infinity..c + I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, z] == Divide[1,2*Pi*I]*Integrate[Gamma[s + a]*Divide[(z)^(- s),s], {s, c - I*Infinity, c + I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1072320848e-1-.1480251451*I
Test Values: {a = -1.5, c = -1.5, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.2224046553+.6479031822e-1*I
Test Values: {a = -1.5, c = -1.5, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.6.E12 8.6.E12] || [[Item:Q2526|<math>\incGamma@{a}{z} = -\frac{z^{a-1}e^{-z}}{\EulerGamma@{1-a}}\*\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\EulerGamma@{s+1-a}\frac{\pi z^{-s}}{\sin@{\pi s}}\diff{s}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{z} = -\frac{z^{a-1}e^{-z}}{\EulerGamma@{1-a}}\*\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\EulerGamma@{s+1-a}\frac{\pi z^{-s}}{\sin@{\pi s}}\diff{s}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{3}{2}\pi, \realpart@@{s+1-a} > 0, \realpart@@{1-a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, z) = -((z)^(a - 1)* exp(- z))/(GAMMA(1 - a))*(1)/(2*Pi*I)*int(GAMMA(s + 1 - a)*(Pi*(z)^(- s))/(sin(Pi*s)), s = c - I*infinity..c + I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, z] == -Divide[(z)^(a - 1)* Exp[- z],Gamma[1 - a]]*Divide[1,2*Pi*I]*Integrate[Gamma[s + 1 - a]*Divide[Pi*(z)^(- s),Sin[Pi*s]], {s, c - I*Infinity, c + I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [168 / 168]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1072320848e-1-.1480251451*I
Test Values: {a = -1.5, c = -1.5, z = 1/2*3^(1/2)+1/2*I, a = -1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .7867555591e-1+.8824866094*I
Test Values: {a = -1.5, c = -1.5, z = -1/2+1/2*I*3^(1/2), a = -1}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.7.E1 8.7.E1] || [[Item:Q2527|<math>\scincgamma@{a}{z} = e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{a+k+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\scincgamma@{a}{z} = e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{a+k+1}}</syntaxhighlight> || <math>\realpart@@{a+k+1} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Successful || Missing Macro Error || - || -
|-
| [https://dlmf.nist.gov/8.7.E1 8.7.E1] || [[Item:Q2527|<math>e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{a+k+1}} = \frac{1}{\EulerGamma@{a}}\sum_{k=0}^{\infty}\frac{(-z)^{k}}{k!(a+k)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{a+k+1}} = \frac{1}{\EulerGamma@{a}}\sum_{k=0}^{\infty}\frac{(-z)^{k}}{k!(a+k)}</syntaxhighlight> || <math>\realpart@@{a+k+1} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..infinity) = (1)/(GAMMA(a))*sum(((- z)^(k))/(factorial(k)*(a + k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[- z]*Sum[Divide[(z)^(k),Gamma[a + k + 1]], {k, 0, Infinity}, GenerateConditions->None] == Divide[1,Gamma[a]]*Sum[Divide[(- z)^(k),(k)!*(a + k)], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.7.E2 8.7.E2] || [[Item:Q2528|<math>\incgamma@{a}{x+y}-\incgamma@{a}{x} = \incGamma@{a}{x}-\incGamma@{a}{x+y}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{x+y}-\incgamma@{a}{x} = \incGamma@{a}{x}-\incGamma@{a}{x+y}</syntaxhighlight> || <math>|y| < |x|, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, x + y)- GAMMA(a)-GAMMA(a, x) = GAMMA(a, x)- GAMMA(a, x + y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, x + y]- Gamma[a, 0, x] == Gamma[a, x]- Gamma[a, x + y]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.6941375518
Test Values: {a = 1.5, x = 1.5, y = -.5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.6941375518
Test Values: {a = 1.5, x = 1.5, y = .5}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 18]
|-
| [https://dlmf.nist.gov/8.7.E2 8.7.E2] || [[Item:Q2528|<math>\incGamma@{a}{x}-\incGamma@{a}{x+y} = e^{-x}x^{a-1}\sum_{n=0}^{\infty}\frac{\Pochhammersym{1-a}{n}}{(-x)^{n}}(1-e^{-y}e_{n}(y))</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{x}-\incGamma@{a}{x+y} = e^{-x}x^{a-1}\sum_{n=0}^{\infty}\frac{\Pochhammersym{1-a}{n}}{(-x)^{n}}(1-e^{-y}e_{n}(y))</syntaxhighlight> || <math>|y| < |x|, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, x)- GAMMA(a, x + y) = exp(- x)*(x)^(a - 1)* sum((pochhammer(1 - a, n))/((- x)^(n))*(1 - exp(- y)*exp(1)[n]*(y)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, x]- Gamma[a, x + y] == Exp[- x]*(x)^(a - 1)* Sum[Divide[Pochhammer[1 - a, n],(- x)^(n)]*(1 - Exp[- y]*Subscript[E, n]*(y)), {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || Skip - symbolical successful subtest || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.7.E3 8.7.E3] || [[Item:Q2529|<math>\incGamma@{a}{z} = \EulerGamma@{a}-\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{a+k}}{k!(a+k)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{z} = \EulerGamma@{a}-\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{a+k}}{k!(a+k)}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, z) = GAMMA(a)- sum(((- 1)^(k)* (z)^(a + k))/(factorial(k)*(a + k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, z] == Gamma[a]- Sum[Divide[(- 1)^(k)* (z)^(a + k),(k)!*(a + k)], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/8.7.E3 8.7.E3] || [[Item:Q2529|<math>\EulerGamma@{a}-\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{a+k}}{k!(a+k)} = \EulerGamma@{a}\left(1-z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{a+k+1}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{a}-\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{a+k}}{k!(a+k)} = \EulerGamma@{a}\left(1-z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{a+k+1}}\right)</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{a+k+1} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)- sum(((- 1)^(k)* (z)^(a + k))/(factorial(k)*(a + k)), k = 0..infinity) = GAMMA(a)*(1 - (z)^(a)* exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..infinity))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a]- Sum[Divide[(- 1)^(k)* (z)^(a + k),(k)!*(a + k)], {k, 0, Infinity}, GenerateConditions->None] == Gamma[a]*(1 - (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Gamma[a + k + 1]], {k, 0, Infinity}, GenerateConditions->None])</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/8.7.E6 8.7.E6] || [[Item:Q2532|<math>\incGamma@{a}{x} = x^{a}e^{-x}\sum_{n=0}^{\infty}\frac{\LaguerrepolyL[a]{n}@{x}}{n+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{x} = x^{a}e^{-x}\sum_{n=0}^{\infty}\frac{\LaguerrepolyL[a]{n}@{x}}{n+1}</syntaxhighlight> || <math>x > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, x) = (x)^(a)* exp(- x)*sum((LaguerreL(n, a, x))/(n + 1), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, x] == (x)^(a)* Exp[- x]*Sum[Divide[LaguerreL[n, a, x],n + 1], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.03483445061608508, Times[-0.1214566752420326, NSum[Times[Power[Plus[1, n], -1], LaguerreL[n, -1.5, 1.5]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[a, -1.5], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.7498909754592095, Times[-1.7155277699214138, NSum[Times[Power[Plus[1, n], -1], LaguerreL[n, -1.5, 0.5]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[a, -1.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.8.E1 8.8.E1] || [[Item:Q2533|<math>\incgamma@{a+1}{z} = a\incgamma@{a}{z}-z^{a}e^{-z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a+1}{z} = a\incgamma@{a}{z}-z^{a}e^{-z}</syntaxhighlight> || <math>\realpart@@{a+1} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a + 1)-GAMMA(a + 1, z) = a*GAMMA(a)-GAMMA(a, z)- (z)^(a)* exp(- z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a + 1, 0, z] == a*Gamma[a, 0, z]- (z)^(a)* Exp[- z]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.2676693395+.995081412e-1*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.820738205+.231239721*I
Test Values: {a = 1.5, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.8.E2 8.8.E2] || [[Item:Q2534|<math>\incGamma@{a+1}{z} = a\incGamma@{a}{z}+z^{a}e^{-z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a+1}{z} = a\incGamma@{a}{z}+z^{a}e^{-z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(a + 1, z) = a*GAMMA(a, z)+ (z)^(a)* exp(- z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a + 1, z] == a*Gamma[a, z]+ (z)^(a)* Exp[- z]</syntaxhighlight> || Failure || Successful || Successful [Tested: 42] || Successful [Tested: 42]
|-
| [https://dlmf.nist.gov/8.8.E3 8.8.E3] || [[Item:Q2535|<math>w(a+2,z)-(a+1+z)w(a+1,z)+azw(a,z) = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w(a+2,z)-(a+1+z)w(a+1,z)+azw(a,z) = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>w(a + 2 , z)-(a + 1 + z)*w(a + 1 , z)+ azw(a , z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>w[a + 2 , z]-(a + 1 + z)*w[a + 1 , z]+ azw[a , z] == 0</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.8.E4 8.8.E4] || [[Item:Q2536|<math>z\scincgamma@{a+1}{z} = \scincgamma@{a}{z}-\frac{e^{-z}}{\EulerGamma@{a+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z\scincgamma@{a+1}{z} = \scincgamma@{a}{z}-\frac{e^{-z}}{\EulerGamma@{a+1}}</syntaxhighlight> || <math>\realpart@@{a+1} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>z*(z)^(-(a + 1))*(GAMMA(a + 1)-GAMMA(a + 1, z))/GAMMA(a + 1) = (z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a)-(exp(- z))/(GAMMA(a + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Failure || Missing Macro Error || Successful [Tested: 21] || -
|-
| [https://dlmf.nist.gov/8.8.E5 8.8.E5] || [[Item:Q2537|<math>\normincGammaP@{a+1}{z} = \normincGammaP@{a}{z}-\frac{z^{a}e^{-z}}{\EulerGamma@{a+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincGammaP@{a+1}{z} = \normincGammaP@{a}{z}-\frac{z^{a}e^{-z}}{\EulerGamma@{a+1}}</syntaxhighlight> || <math>\realpart@@{a+1} > 0</math> || <syntaxhighlight lang=mathematica>(GAMMA(a + 1)-GAMMA(a + 1, z))/GAMMA(a + 1) = (GAMMA(a)-GAMMA(a, z))/GAMMA(a)-((z)^(a)* exp(- z))/(GAMMA(a + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>GammaRegularized[a + 1, 0, z] == GammaRegularized[a, 0, z]-Divide[(z)^(a)* Exp[- z],Gamma[a + 1]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 28] || Successful [Tested: 28]
|-
| [https://dlmf.nist.gov/8.8.E6 8.8.E6] || [[Item:Q2538|<math>\normincGammaQ@{a+1}{z} = \normincGammaQ@{a}{z}+\frac{z^{a}e^{-z}}{\EulerGamma@{a+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincGammaQ@{a+1}{z} = \normincGammaQ@{a}{z}+\frac{z^{a}e^{-z}}{\EulerGamma@{a+1}}</syntaxhighlight> || <math>\realpart@@{a+1} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a + 1, z)/GAMMA(a + 1) = GAMMA(a, z)/GAMMA(a)+((z)^(a)* exp(- z))/(GAMMA(a + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>GammaRegularized[a + 1, z] == GammaRegularized[a, z]+Divide[(z)^(a)* Exp[- z],Gamma[a + 1]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 28] || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.8.E7 8.8.E7] || [[Item:Q2539|<math>\incgamma@{a+n}{z} = \Pochhammersym{a}{n}\incgamma@{a}{z}-z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{\EulerGamma@{a+n}}{\EulerGamma@{a+k+1}}z^{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a+n}{z} = \Pochhammersym{a}{n}\incgamma@{a}{z}-z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{\EulerGamma@{a+n}}{\EulerGamma@{a+k+1}}z^{k}</syntaxhighlight> || <math>\realpart@@{a+n} > 0, \realpart@@{a+k+1} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a + n)-GAMMA(a + n, z) = pochhammer(a, n)*GAMMA(a)-GAMMA(a, z)- (z)^(a)* exp(- z)*sum((GAMMA(a + n))/(GAMMA(a + k + 1))*(z)^(k), k = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a + n, 0, z] == Pochhammer[a, n]*Gamma[a, 0, z]- (z)^(a)* Exp[- z]*Sum[Divide[Gamma[a + n],Gamma[a + k + 1]]*(z)^(k), {k, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [63 / 63]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.2676693391+.995081412e-1*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.472181365+.5472947763*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 63]
|-
| [https://dlmf.nist.gov/8.8.E8 8.8.E8] || [[Item:Q2540|<math>\incgamma@{a}{z} = \frac{\EulerGamma@{a}}{\EulerGamma@{a-n}}\incgamma@{a-n}{z}-z^{a-1}e^{-z}\sum_{k=0}^{n-1}\frac{\EulerGamma@{a}}{\EulerGamma@{a-k}}z^{-k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{z} = \frac{\EulerGamma@{a}}{\EulerGamma@{a-n}}\incgamma@{a-n}{z}-z^{a-1}e^{-z}\sum_{k=0}^{n-1}\frac{\EulerGamma@{a}}{\EulerGamma@{a-k}}z^{-k}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{a-n} > 0, \realpart@@{a-k} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, z) = (GAMMA(a))/(GAMMA(a - n))*GAMMA(a - n)-GAMMA(a - n, z)- (z)^(a - 1)* exp(- z)*sum((GAMMA(a))/(GAMMA(a - k))*(z)^(- k), k = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, z] == Divide[Gamma[a],Gamma[a - n]]*Gamma[a - n, 0, z]- (z)^(a - 1)* Exp[- z]*Sum[Divide[Gamma[a],Gamma[a - k]]*(z)^(- k), {k, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 14]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1265952281-.9976912441e-1*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .19739482e-1-.7595504274*I
Test Values: {a = 1.5, z = -1/2+1/2*I*3^(1/2), n = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 14]
|-
| [https://dlmf.nist.gov/8.8.E9 8.8.E9] || [[Item:Q2541|<math>\incGamma@{a+n}{z} = \Pochhammersym{a}{n}\incGamma@{a}{z}+z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{\EulerGamma@{a+n}}{\EulerGamma@{a+k+1}}z^{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a+n}{z} = \Pochhammersym{a}{n}\incGamma@{a}{z}+z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{\EulerGamma@{a+n}}{\EulerGamma@{a+k+1}}z^{k}</syntaxhighlight> || <math>\realpart@@{a+n} > 0, \realpart@@{a+k+1} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a + n, z) = pochhammer(a, n)*GAMMA(a, z)+ (z)^(a)* exp(- z)*sum((GAMMA(a + n))/(GAMMA(a + k + 1))*(z)^(k), k = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a + n, z] == Pochhammer[a, n]*Gamma[a, z]+ (z)^(a)* Exp[- z]*Sum[Divide[Gamma[a + n],Gamma[a + k + 1]]*(z)^(k), {k, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 105]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.8.E10 8.8.E10] || [[Item:Q2542|<math>\incGamma@{a}{z} = \frac{\EulerGamma@{a}}{\EulerGamma@{a-n}}\incGamma@{a-n}{z}+z^{a-1}e^{-z}\sum_{k=0}^{n-1}\frac{\EulerGamma@{a}}{\EulerGamma@{a-k}}z^{-k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{z} = \frac{\EulerGamma@{a}}{\EulerGamma@{a-n}}\incGamma@{a-n}{z}+z^{a-1}e^{-z}\sum_{k=0}^{n-1}\frac{\EulerGamma@{a}}{\EulerGamma@{a-k}}z^{-k}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{a-n} > 0, \realpart@@{a-k} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, z) = (GAMMA(a))/(GAMMA(a - n))*GAMMA(a - n, z)+ (z)^(a - 1)* exp(- z)*sum((GAMMA(a))/(GAMMA(a - k))*(z)^(- k), k = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, z] == Divide[Gamma[a],Gamma[a - n]]*Gamma[a - n, z]+ (z)^(a - 1)* Exp[- z]*Sum[Divide[Gamma[a],Gamma[a - k]]*(z)^(- k), {k, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 14] || Successful [Tested: 14]
|-
| [https://dlmf.nist.gov/8.8.E11 8.8.E11] || [[Item:Q2543|<math>\normincGammaP@{a+n}{z} = \normincGammaP@{a}{z}-z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{\EulerGamma@{a+k+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincGammaP@{a+n}{z} = \normincGammaP@{a}{z}-z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{\EulerGamma@{a+k+1}}</syntaxhighlight> || <math>\realpart@@{a+k+1} > 0</math> || <syntaxhighlight lang=mathematica>(GAMMA(a + n)-GAMMA(a + n, z))/GAMMA(a + n) = (GAMMA(a)-GAMMA(a, z))/GAMMA(a)- (z)^(a)* exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>GammaRegularized[a + n, 0, z] == GammaRegularized[a, 0, z]- (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Gamma[a + k + 1]], {k, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -2], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -2], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.8.E12 8.8.E12] || [[Item:Q2544|<math>\normincGammaQ@{a+n}{z} = \normincGammaQ@{a}{z}+z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{\EulerGamma@{a+k+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincGammaQ@{a+n}{z} = \normincGammaQ@{a}{z}+z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{\EulerGamma@{a+k+1}}</syntaxhighlight> || <math>\realpart@@{a+k+1} > 0, \realpart@@{a+n} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a + n, z)/GAMMA(a + n) = GAMMA(a, z)/GAMMA(a)+ (z)^(a)* exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>GammaRegularized[a + n, z] == GammaRegularized[a, z]+ (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Gamma[a + k + 1]], {k, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 63]
|-
| [https://dlmf.nist.gov/8.8.E13 8.8.E13] || [[Item:Q2545|<math>\deriv{}{z}\incgamma@{a}{z} = -\deriv{}{z}\incGamma@{a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{z}\incgamma@{a}{z} = -\deriv{}{z}\incGamma@{a}{z}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>diff(GAMMA(a)-GAMMA(a, z), z) = - diff(GAMMA(a, z), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Gamma[a, 0, z], z] == - D[Gamma[a, z], z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.8.E13 8.8.E13] || [[Item:Q2545|<math>-\deriv{}{z}\incGamma@{a}{z} = z^{a-1}e^{-z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\deriv{}{z}\incGamma@{a}{z} = z^{a-1}e^{-z}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>- diff(GAMMA(a, z), z) = (z)^(a - 1)* exp(- z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- D[Gamma[a, z], z] == (z)^(a - 1)* Exp[- z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.8.E15 8.8.E15] || [[Item:Q2547|<math>\deriv[n]{}{z}(z^{-a}\incgamma@{a}{z}) = (-1)^{n}z^{-a-n}\incgamma@{a+n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[n]{}{z}(z^{-a}\incgamma@{a}{z}) = (-1)^{n}z^{-a-n}\incgamma@{a+n}{z}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{a+n} > 0</math> || <syntaxhighlight lang=mathematica>diff((z)^(- a)* GAMMA(a)-GAMMA(a, z), [z$(n)]) = (- 1)^(n)* (z)^(- a - n)* GAMMA(a + n)-GAMMA(a + n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[(z)^(- a)* Gamma[a, 0, z], {z, n}] == (- 1)^(n)* (z)^(- a - n)* Gamma[a + n, 0, z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [63 / 63]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.615357258-.2793504168*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.050292670-.1918135000*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [63 / 63]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.20573036539123668, -0.07193062032175179], Inactive[Sum][Times[Power[Complex[0.8660254037844387, 0.49999999999999994], Plus[-1.5, Times[-1.0, K[1.0]]]], Binomial[1.0, K[1.0]], D[Complex[0.3508882473022298, 0.19901628242832037]
Test Values: {Complex[0.8660254037844387, 0.49999999999999994], Plus[1.0, Times[-1.0, K[1.0]]]}], FactorialPower[-1.5, K[1.0]]], {K[1.0], 0.0, 1.0}]], {Rule[a, 1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.13665910465469025, 0.05369371428345661], Inactive[Sum][Times[Power[Complex[0.8660254037844387, 0.49999999999999994], Plus[-1.5, Times[-1.0, K[1.0]]]], Binomial[2.0, K[1.0]], D[Complex[0.3508882473022298, 0.19901628242832037]
Test Values: {Complex[0.8660254037844387, 0.49999999999999994], Plus[2.0, Times[-1.0, K[1.0]]]}], FactorialPower[-1.5, K[1.0]]], {K[1.0], 0.0, 2.0}]], {Rule[a, 1.5], Rule[n, 2],</div></div>
|-
| [https://dlmf.nist.gov/8.8.E16 8.8.E16] || [[Item:Q2548|<math>\deriv[n]{}{z}(z^{-a}\incGamma@{a}{z}) = (-1)^{n}z^{-a-n}\incGamma@{a+n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[n]{}{z}(z^{-a}\incGamma@{a}{z}) = (-1)^{n}z^{-a-n}\incGamma@{a+n}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff((z)^(- a)* GAMMA(a, z), [z$(n)]) = (- 1)^(n)* (z)^(- a - n)* GAMMA(a + n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[(z)^(- a)* Gamma[a, z], {z, n}] == (- 1)^(n)* (z)^(- a - n)* Gamma[a + n, z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [111 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.14584260074790834, -0.14889469354125948], Times[Complex[0.9659258262890683, 0.25881904510252074], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, , Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], []], Times[-1, Plus[-1, Times[-1, ], 1], Plus[1, Times[3, ], Times[3, Power[, 2]], Power[, 3], Times[-2, -1.5], Times[-4, , -1.5], Times[-2, Power[, 2], -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[-1, 1], Times[-2, , 1], Times[-1, Power[, 2], 1], Times[-1.5, 1], Times[, -1.5, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Times[2, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, Power[, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , -1.5, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 1, Power[E, Times[Complex[0, Rational[1, 6]], </div></div>
|-
| [https://dlmf.nist.gov/8.8.E17 8.8.E17] || [[Item:Q2549|<math>\deriv[n]{}{z}(e^{z}\incgamma@{a}{z}) = (-1)^{n}\Pochhammersym{1-a}{n}e^{z}\incgamma@{a-n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[n]{}{z}(e^{z}\incgamma@{a}{z}) = (-1)^{n}\Pochhammersym{1-a}{n}e^{z}\incgamma@{a-n}{z}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{a-n} > 0</math> || <syntaxhighlight lang=mathematica>diff(exp(z)*GAMMA(a)-GAMMA(a, z), [z$(n)]) = (- 1)^(n)* pochhammer(1 - a, n)*exp(z)*GAMMA(a - n)-GAMMA(a - n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Exp[z]*Gamma[a, 0, z], {z, n}] == (- 1)^(n)* Pochhammer[1 - a, n]*Exp[z]*Gamma[a - n, 0, z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [14 / 14]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .6619339064-.2987854069*I
Test Values: {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.661215891-1.222029870*I
Test Values: {a = 1.5, z = -1/2+1/2*I*3^(1/2), n = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [14 / 14]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-1.471179750131411, -1.0739918488339026], Inactive[Sum][Times[Complex[2.0864022336812553, 1.1398067350757155], Binomial[1.0, K[1.0]], D[Complex[0.3508882473022298, 0.19901628242832037]
Test Values: {Complex[0.8660254037844387, 0.49999999999999994], Plus[1.0, Times[-1.0, K[1.0]]]}]], {K[1.0], 0.0, 1.0}]], {Rule[a, 1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.01045242262446705, -0.698806597134537], Inactive[Sum][Times[Complex[0.3929465558343552, 0.4620307840711054], Binomial[1.0, K[1.0]], D[Complex[-0.7552494829576352, 0.46247944264186114]
Test Values: {Complex[-0.4999999999999998, 0.8660254037844387], Plus[1.0, Times[-1.0, K[1.0]]]}]], {K[1.0], 0.0, 1.0}]], {Rule[a, 1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.8.E18 8.8.E18] || [[Item:Q2550|<math>\deriv[n]{}{z}(z^{a}e^{z}\scincgamma@{a}{z}) = z^{a-n}e^{z}\scincgamma@{a-n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[n]{}{z}(z^{a}e^{z}\scincgamma@{a}{z}) = z^{a-n}e^{z}\scincgamma@{a-n}{z}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{a-n} > 0</math> || <syntaxhighlight lang=mathematica>diff((z)^(a)* exp(z)*(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a), [z$(n)]) = (z)^(a - n)* exp(z)*(z)^(-(a - n))*(GAMMA(a - n)-GAMMA(a - n, z))/GAMMA(a - n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Failure || Missing Macro Error || Successful [Tested: 14] || -
|-
| [https://dlmf.nist.gov/8.8.E19 8.8.E19] || [[Item:Q2551|<math>\deriv[n]{}{z}(e^{z}\incGamma@{a}{z}) = (-1)^{n}\Pochhammersym{1-a}{n}e^{z}\incGamma@{a-n}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[n]{}{z}(e^{z}\incGamma@{a}{z}) = (-1)^{n}\Pochhammersym{1-a}{n}e^{z}\incGamma@{a-n}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(exp(z)*GAMMA(a, z), [z$(n)]) = (- 1)^(n)* pochhammer(1 - a, n)*exp(z)*GAMMA(a - n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Exp[z]*Gamma[a, z], {z, n}] == (- 1)^(n)* Pochhammer[1 - a, n]*Exp[z]*Gamma[a - n, z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [96 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.06772606154573046, -0.6693082179083164], Times[Complex[2.0864022336812553, 1.1398067350757155], DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[-1, , Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], []], Times[Plus[-1, Times[-1, ], 1], Plus[-1, Times[-2, ], Times[-2, Power[, 2]], -1.5, Times[, -1.5], Times[, 1], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[1, ]]], Times[-1, Plus[1, ], Plus[1, Times[2, ], Power[, 2], Times[-1, -1.5], Times[-1, , -1.5], Times[-1, 1], Times[-1, , 1], Times[-1.5, 1], Times[3, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[2, , Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[-1, 1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1</div></div>
|-
| [https://dlmf.nist.gov/8.10.E1 8.10.E1] || [[Item:Q2554|<math>x^{1-a}e^{x}\incGamma@{a}{x} \leq 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x^{1-a}e^{x}\incGamma@{a}{x} \leq 1</syntaxhighlight> || <math>x > 0, 0 < a, a \leq 1</math> || <syntaxhighlight lang=mathematica>(x)^(1 - a)* exp(x)*GAMMA(a, x) <= 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>(x)^(1 - a)* Exp[x]*Gamma[a, x] <= 1</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/8.10.E2 8.10.E2] || [[Item:Q2555|<math>\incgamma@{a}{x} \geq \frac{x^{a-1}}{a}(1-e^{-x})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{x} \geq \frac{x^{a-1}}{a}(1-e^{-x})</syntaxhighlight> || <math>x > 0, 0 < a, a \leq 1, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, x) >= ((x)^(a - 1))/(a)*(1 - exp(- x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, x] >= Divide[(x)^(a - 1),a]*(1 - Exp[- x])</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/8.10.E3 8.10.E3] || [[Item:Q2556|<math>x^{1-a}e^{x}\incGamma@{a}{x} = 1+\frac{a-1}{x}\vartheta</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x^{1-a}e^{x}\incGamma@{a}{x} = 1+\frac{a-1}{x}\vartheta</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(x)^(1 - a)* exp(x)*GAMMA(a, x) = 1 +(a - 1)/(x)*vartheta</syntaxhighlight> || <syntaxhighlight lang=mathematica>(x)^(1 - a)* Exp[x]*Gamma[a, x] == 1 +Divide[a - 1,x]*\[CurlyTheta]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8735840245+.8333333335*I
Test Values: {a = -1.5, x = 1.5, vartheta = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.403124983+1.443375674*I
Test Values: {a = -1.5, x = 1.5, vartheta = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.8735840235649492, 0.8333333333333331]
Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[ϑ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.4031249827424481, 1.4433756729740643]
Test Values: {Rule[a, -1.5], Rule[x, 1.5], Rule[ϑ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.10.E4 8.10.E4] || [[Item:Q2557|<math>0 < \vartheta</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>0 < \vartheta</syntaxhighlight> || <math>x > 0, a \leq 2</math> || <syntaxhighlight lang=mathematica>0 < vartheta</syntaxhighlight> || <syntaxhighlight lang=mathematica>0 < \[CurlyTheta]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.10.E5 8.10.E5] || [[Item:Q2558|<math>A_{n} < x^{1-a}e^{x}\incGamma@{a}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>A_{n} < x^{1-a}e^{x}\incGamma@{a}{x}</syntaxhighlight> || <math>x > 0, a < 1</math> || <syntaxhighlight lang=mathematica>A[n](<)*(x)^(1 - a)* exp(x)*GAMMA(a, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[A, n][<]*(x)^(1 - a)* Exp[x]*Gamma[a, x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [75 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.5 < .4302083505
Test Values: {a = -1.5, x = 1.5, A[n] = 1.5, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.5 < .4302083505
Test Values: {a = -1.5, x = 1.5, A[n] = 1.5, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [195 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Less[Complex[0.8660254037844387, 0.49999999999999994], 0.43020835059088497]
Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[Subscript[A, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Less[Complex[0.8660254037844387, 0.49999999999999994], 0.43020835059088497]
Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[x, 1.5], Rule[Subscript[A, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.10.E5 8.10.E5] || [[Item:Q2558|<math>x^{1-a}e^{x}\incGamma@{a}{x} < B_{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x^{1-a}e^{x}\incGamma@{a}{x} < B_{n}</syntaxhighlight> || <math>x > 0, a < 1</math> || <syntaxhighlight lang=mathematica>(x)^(1 - a)* exp(x)*GAMMA(a, x) < B[n]</syntaxhighlight> || <syntaxhighlight lang=mathematica>(x)^(1 - a)* Exp[x]*Gamma[a, x] < Subscript[B, n]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [105 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .4302083505 < -1.5
Test Values: {a = -1.5, x = 1.5, B[n] = -1.5, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .4302083505 < -1.5
Test Values: {a = -1.5, x = 1.5, B[n] = -1.5, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [225 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Less[0.43020835059088497, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[Subscript[B, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Less[0.43020835059088497, Complex[0.8660254037844387, 0.49999999999999994]]
Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[x, 1.5], Rule[Subscript[B, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.10#Ex1 8.10#Ex1] || [[Item:Q2559|<math>A_{1} = \frac{x}{x+1-a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>A_{1} = \frac{x}{x+1-a}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>A[1] = (x)/(x + 1 - a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[A, 1] == Divide[x,x + 1 - a]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.10#Ex2 8.10#Ex2] || [[Item:Q2560|<math>B_{1} = \frac{x+1}{x+2-a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>B_{1} = \frac{x+1}{x+2-a}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>B[1] = (x + 1)/(x + 2 - a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[B, 1] == Divide[x + 1,x + 2 - a]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.10#Ex3 8.10#Ex3] || [[Item:Q2561|<math>A_{2} = \frac{x(x+3-a)}{x^{2}+2(2-a)x+(1-a)(2-a)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>A_{2} = \frac{x(x+3-a)}{x^{2}+2(2-a)x+(1-a)(2-a)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>A[2] = (x*(x + 3 - a))/((x)^(2)+ 2*(2 - a)*x +(1 - a)*(2 - a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[A, 2] == Divide[x*(x + 3 - a),(x)^(2)+ 2*(2 - a)*x +(1 - a)*(2 - a)]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.10#Ex4 8.10#Ex4] || [[Item:Q2562|<math>B_{2} = \frac{x^{2}+(5-a)x+2}{x^{2}+2(3-a)x+(2-a)(3-a)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>B_{2} = \frac{x^{2}+(5-a)x+2}{x^{2}+2(3-a)x+(2-a)(3-a)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>B[2] = ((x)^(2)+(5 - a)*x + 2)/((x)^(2)+ 2*(3 - a)*x +(2 - a)*(3 - a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[B, 2] == Divide[(x)^(2)+(5 - a)*x + 2,(x)^(2)+ 2*(3 - a)*x +(2 - a)*(3 - a)]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.10.E7 8.10.E7] || [[Item:Q2563|<math>I = \int_{0}^{x}t^{a-1}e^{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>I = \int_{0}^{x}t^{a-1}e^{t}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>I = int((t)^(a - 1)* exp(t), t = 0..x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>I == Integrate[(t)^(a - 1)* Exp[t], {t, 0, x}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [90 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -2.374226751+.5000000000*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, a = 1.5, x = 1.5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .5451660792+.5000000000*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, a = 1.5, x = .5}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [27 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-2.240252154794788, -4.0113152157384396*^-17]
Test Values: {Rule[Complex[0, 1], 1], Rule[a, 1.5], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.240252154794788, -4.0113152157384396*^-17]
Test Values: {Rule[Complex[0, 1], 2], Rule[a, 1.5], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.10.E7 8.10.E7] || [[Item:Q2563|<math>\int_{0}^{x}t^{a-1}e^{t}\diff{t} = \EulerGamma@{a}x^{a}\scincgamma@{a}{-x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{x}t^{a-1}e^{t}\diff{t} = \EulerGamma@{a}x^{a}\scincgamma@{a}{-x}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^(a - 1)* exp(t), t = 0..x) = GAMMA(a)*(x)^(a)* (- x)^(-(a))*(GAMMA(a)-GAMMA(a, - x))/GAMMA(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Failure || Missing Macro Error || Successful [Tested: 9] || Skip - symbolical successful subtest
|-
| [https://dlmf.nist.gov/8.10.E8 8.10.E8] || [[Item:Q2564|<math>\frac{(a+1)(a+2)-x}{(a+1)(a+2+x)} < ax^{-a}e^{-x}I</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{(a+1)(a+2)-x}{(a+1)(a+2+x)} < ax^{-a}e^{-x}I</syntaxhighlight> || <math>x > 0, a \geq 0</math> || <syntaxhighlight lang=mathematica>((a + 1)*(a + 2)- x)/((a + 1)*(a + 2 + x)) < a*(x)^(- a)* exp(- x)*I</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[(a + 1)*(a + 2)- x,(a + 1)*(a + 2 + x)] < a*(x)^(- a)* Exp[- x]*I</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.10#Ex5 8.10#Ex5] || [[Item:Q2565|<math>c_{a} = (\EulerGamma@{1+a})^{1/(a-1)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c_{a} = (\EulerGamma@{1+a})^{1/(a-1)}</syntaxhighlight> || <math>\realpart@@{1+a} > 0</math> || <syntaxhighlight lang=mathematica>c[a] = (GAMMA(1 + a))^(1/(a - 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[c, a] == (Gamma[1 + a])^(1/(a - 1))</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [39 / 40]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.9011204630+.5000000000*I
Test Values: {a = 1.5, c[a] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -2.267145867+.8660254040*I
Test Values: {a = 1.5, c[a] = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [39 / 40]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.9011204638598199, 0.49999999999999994]
Test Values: {Rule[a, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-2.2671458676442584, 0.8660254037844387]
Test Values: {Rule[a, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.10#Ex6 8.10#Ex6] || [[Item:Q2566|<math>d_{a} = (\EulerGamma@{1+a})^{-1/a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>d_{a} = (\EulerGamma@{1+a})^{-1/a}</syntaxhighlight> || <math>\realpart@@{1+a} > 0</math> || <syntaxhighlight lang=mathematica>d[a] = (GAMMA(1 + a))^(- 1/a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[d, a] == (Gamma[1 + a])^(- 1/a)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [40 / 40]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .388914161e-1+.5000000000*I
Test Values: {a = 1.5, d[a] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.327133988+.8660254040*I
Test Values: {a = 1.5, d[a] = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [40 / 40]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.038891415918572037, 0.49999999999999994]
Test Values: {Rule[a, 1.5], Rule[Subscript[d, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.3271339878658663, 0.8660254037844387]
Test Values: {Rule[a, 1.5], Rule[Subscript[d, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.10.E10 8.10.E10] || [[Item:Q2567|<math>\frac{x}{2a}\left(\left(1+\frac{2}{x}\right)^{a}-1\right) < x^{1-a}e^{x}\incGamma@{a}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{x}{2a}\left(\left(1+\frac{2}{x}\right)^{a}-1\right) < x^{1-a}e^{x}\incGamma@{a}{x}</syntaxhighlight> || <math>x \geq 0, 0 < a, a < 1</math> || <syntaxhighlight lang=mathematica>(x)/(2*a)*((1 +(2)/(x))^(a)- 1) < (x)^(1 - a)* exp(x)*GAMMA(a, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[x,2*a]*((1 +Divide[2,x])^(a)- 1) < (x)^(1 - a)* Exp[x]*Gamma[a, x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/8.10.E10 8.10.E10] || [[Item:Q2567|<math>x^{1-a}e^{x}\incGamma@{a}{x} \leq \frac{x}{ac_{a}}\left(\left(1+\frac{c_{a}}{x}\right)^{a}-1\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x^{1-a}e^{x}\incGamma@{a}{x} \leq \frac{x}{ac_{a}}\left(\left(1+\frac{c_{a}}{x}\right)^{a}-1\right)</syntaxhighlight> || <math>x \geq 0, 0 < a, a < 1</math> || <syntaxhighlight lang=mathematica>(x)^(1 - a)* exp(x)*GAMMA(a, x) <= (x)/(a*c[a])*((1 +(c[a])/(x))^(a)- 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(x)^(1 - a)* Exp[x]*Gamma[a, x] <= Divide[x,a*Subscript[c, a]]*((1 +Divide[Subscript[c, a],x])^(a)- 1)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8100694499 <= .7912878480
Test Values: {a = .5, x = 1.5, c[a] = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .6556795425 <= .6180339885
Test Values: {a = .5, x = .5, c[a] = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[0.8100694501969615, Complex[0.8808481919138387, -0.05137441674828974]]
Test Values: {Rule[a, 0.5], Rule[x, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[0.8100694501969615, Complex[1.0324243733930456, -0.1801654927820326]]
Test Values: {Rule[a, 0.5], Rule[x, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.10.E11 8.10.E11] || [[Item:Q2568|<math>(1-e^{-\alpha_{a}x})^{a} \leq \normincGammaP@{a}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(1-e^{-\alpha_{a}x})^{a} \leq \normincGammaP@{a}{x}</syntaxhighlight> || <math>x \geq 0, a > 0</math> || <syntaxhighlight lang=mathematica>(1 - exp(- alpha[a]*x))^(a) <= (GAMMA(a)-GAMMA(a, x))/GAMMA(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(1 - Exp[- Subscript[\[Alpha], a]*x])^(a) <= GammaRegularized[a, 0, x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [78 / 270]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8461432717 <= .6083748237
Test Values: {a = 1.5, alpha = 1.5, x = 1.5, alpha[a] = 1.5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .9262567903 <= .6083748237
Test Values: {a = 1.5, alpha = 1.5, x = 1.5, alpha[a] = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [240 / 270]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[0.7016331692747775, 0.2500919864059583], 0.608374823728911]
Test Values: {Rule[a, 1.5], Rule[x, 1.5], Rule[α, 1.5], Rule[Subscript[α, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[Complex[-1.369574242346028, 2.679891945423719], 0.608374823728911]
Test Values: {Rule[a, 1.5], Rule[x, 1.5], Rule[α, 1.5], Rule[Subscript[α, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.10.E11 8.10.E11] || [[Item:Q2568|<math>\normincGammaP@{a}{x} \leq (1-e^{-\beta_{a}x})^{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincGammaP@{a}{x} \leq (1-e^{-\beta_{a}x})^{a}</syntaxhighlight> || <math>x \geq 0, a > 0</math> || <syntaxhighlight lang=mathematica>(GAMMA(a)-GAMMA(a, x))/GAMMA(a) <= (1 - exp(- beta[a]*x))^(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>GammaRegularized[a, 0, x] <= (1 - Exp[- Subscript[\[Beta], a]*x])^(a)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 270]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .6083748237 <= .3832643966
Test Values: {a = 1.5, beta = 1.5, x = 1.5, beta[a] = .5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .1987480431 <= .1040340193
Test Values: {a = 1.5, beta = 1.5, x = .5, beta[a] = .5}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [192 / 270]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: LessEqual[0.608374823728911, Complex[0.7016331692747775, 0.2500919864059583]]
Test Values: {Rule[a, 1.5], Rule[x, 1.5], Rule[β, 1.5], Rule[Subscript[β, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: LessEqual[0.608374823728911, Complex[-1.369574242346028, 2.679891945423719]]
Test Values: {Rule[a, 1.5], Rule[x, 1.5], Rule[β, 1.5], Rule[Subscript[β, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.10.E13 8.10.E13] || [[Item:Q2571|<math>\frac{\incGamma@{n}{n}}{\EulerGamma@{n}} < \frac{1}{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\incGamma@{n}{n}}{\EulerGamma@{n}} < \frac{1}{2}</syntaxhighlight> || <math>\realpart@@{n} > 0</math> || <syntaxhighlight lang=mathematica>(GAMMA(n, n))/(GAMMA(n)) < (1)/(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Gamma[n, n],Gamma[n]] < Divide[1,2]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/8.10.E13 8.10.E13] || [[Item:Q2571|<math>\frac{1}{2} < \frac{\incGamma@{n}{n-1}}{\EulerGamma@{n}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{2} < \frac{\incGamma@{n}{n-1}}{\EulerGamma@{n}}</syntaxhighlight> || <math>\realpart@@{n} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(2) < (GAMMA(n, n - 1))/(GAMMA(n))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2] < Divide[Gamma[n, n - 1],Gamma[n]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/8.11.E2 8.11.E2] || [[Item:Q2573|<math>\incGamma@{a}{z} = z^{a-1}e^{-z}\left(\sum_{k=0}^{n-1}\frac{u_{k}}{z^{k}}+R_{n}(a,z)\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incGamma@{a}{z} = z^{a-1}e^{-z}\left(\sum_{k=0}^{n-1}\frac{u_{k}}{z^{k}}+R_{n}(a,z)\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(a, z) = (z)^(a - 1)* exp(- z)*(sum(((- 1)^(k)* pochhammer(1 - a, k))/((z)^(k)), k = 0..n - 1)+ R[n](a , z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, z] == (z)^(a - 1)* Exp[- z]*(Sum[Divide[(- 1)^(k)* Pochhammer[1 - a, k],(z)^(k)], {k, 0, n - 1}, GenerateConditions->None]+ Subscript[R, n][a , z])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1072320848e-1-.1480251451*I+(.9924715785e-1+.4087434498*I)*(1.000000000+0.*I+(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I))
Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, n = 1, n = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .1072320848e-1-.1480251451*I+(.9924715785e-1+.4087434498*I)*(-1.165063509+1.250000000*I+(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I))
Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, n = 2, n = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
|-
| [https://dlmf.nist.gov/8.11.E4 8.11.E4] || [[Item:Q2575|<math>\incgamma@{a}{z} = z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Pochhammersym{a}{k+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{z} = z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Pochhammersym{a}{k+1}}</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, z) = (z)^(a)* exp(- z)*sum(((z)^(k))/(pochhammer(a, k + 1)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, z] == (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Pochhammer[a, k + 1]], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/8.11#Ex1 8.11#Ex1] || [[Item:Q2579|<math>b_{0}(\lambda) = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>b_{0}(\lambda) = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>b[0](lambda) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[b, 0][\[Lambda]] == 1</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.11#Ex2 8.11#Ex2] || [[Item:Q2580|<math>b_{1}(\lambda) = \lambda</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>b_{1}(\lambda) = \lambda</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>b[1](lambda) = lambda</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[b, 1][\[Lambda]] == \[Lambda]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.11#Ex3 8.11#Ex3] || [[Item:Q2581|<math>b_{2}(\lambda) = \lambda(2\lambda+1)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>b_{2}(\lambda) = \lambda(2\lambda+1)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>b[2](lambda) = lambda*(2*lambda + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[b, 2][\[Lambda]] == \[Lambda]*(2*\[Lambda]+ 1)</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.11.E15 8.11.E15] || [[Item:Q2588|<math>S_{n}(x) = \frac{\incgamma@{n+1}{nx}}{(nx)^{n}e^{-nx}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>S_{n}(x) = \frac{\incgamma@{n+1}{nx}}{(nx)^{n}e^{-nx}}</syntaxhighlight> || <math>\realpart@@{n+1} > 0</math> || <syntaxhighlight lang=mathematica>S[n](x) = (GAMMA(n + 1)-GAMMA(n + 1, n*x))/((n*x)^(n)* exp(- n*x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[S, n][x] == Divide[Gamma[n + 1, 0, n*x],(n*x)^(n)* Exp[- n*x]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [90 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.22087941e-1+.7500000000*I
Test Values: {x = 1.5, S[n] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.275525655+.7500000000*I
Test Values: {x = 1.5, S[n] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [90 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.02208794121538471, 0.7499999999999999]
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[Subscript[S, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.2755256550317124, 0.7499999999999999]
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[Subscript[S, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.11.E19 8.11.E19] || [[Item:Q2592|<math>d_{k}(x) = \frac{(-1)^{k}b_{k}(x)}{(1-x)^{2k+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>d_{k}(x) = \frac{(-1)^{k}b_{k}(x)}{(1-x)^{2k+1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>d[k](x) = ((- 1)^(k)* b[k](x))/((1 - x)^(2*k + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[d, k][x] == Divide[(- 1)^(k)* Subscript[b, k][x],(1 - x)^(2*k + 1)]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex1 8.12#Ex1] || [[Item:Q2593|<math>\lambda = z/a</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\lambda = z/a</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>lambda = z/a</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Lambda] == z/a</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex2 8.12#Ex2] || [[Item:Q2594|<math>\eta = \left(2(\lambda-1-\ln@@{\lambda})\right)^{1/2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\eta = \left(2(\lambda-1-\ln@@{\lambda})\right)^{1/2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>eta = (2*(lambda - 1 - ln(lambda)))^(1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Eta] == (2*(\[Lambda]- 1 - Log[\[Lambda]]))^(1/2)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8206105237+1.019626504*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2036159778+2.354396465*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, lambda = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.8206105232686428, 1.019626504138681]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.20361597732323333, 2.3543964646926674]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.12#Ex3 8.12#Ex3] || [[Item:Q2595|<math>\tfrac{1}{2}\eta^{2} = \lambda-1-\ln@@{\lambda}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tfrac{1}{2}\eta^{2} = \lambda-1-\ln@@{\lambda}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(2)*(eta)^(2) = lambda - 1 - ln(lambda)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2]*\[Eta]^(2) == \[Lambda]- 1 - Log[\[Lambda]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .3839745964+.4566114775*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.750000000+1.661382400*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, lambda = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.38397459621556135, 0.45661147749051817]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.7499999999999998, 1.6613824005009756]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.12#Ex4 8.12#Ex4] || [[Item:Q2596|<math>\deriv{\eta}{\lambda} = \frac{\lambda-1}{\lambda\eta}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{\eta}{\lambda} = \frac{\lambda-1}{\lambda\eta}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(eta, lambda) = (lambda - 1)/(lambda*eta)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[\[Eta], \[Lambda]] == Divide[\[Lambda]- 1,\[Lambda]*\[Eta]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.3660254037-.3660254035*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.732050807-.2266367838e-9*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, lambda = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.3660254037844386, -0.3660254037844386]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.7320508075688772, 3.3306690738754696*^-16]
Test Values: {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.12.E3 8.12.E3] || [[Item:Q2597|<math>\normincGammaP@{a}{z} = \tfrac{1}{2}\erfc@{-\eta\sqrt{a/2}}-S(a,\eta)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincGammaP@{a}{z} = \tfrac{1}{2}\erfc@{-\eta\sqrt{a/2}}-S(a,\eta)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = (1)/(2)*erfc(- eta*sqrt(a/2))- S(a , eta)</syntaxhighlight> || <syntaxhighlight lang=mathematica>GammaRegularized[a, 0, z] == Divide[1,2]*Erfc[- \[Eta]*Sqrt[a/2]]- S[a , \[Eta]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8724483635-.3325384943*I+(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {S = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8436948583-.7685914925*I+(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {S = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
|-
| [https://dlmf.nist.gov/8.12.E4 8.12.E4] || [[Item:Q2598|<math>\normincGammaQ@{a}{z} = \tfrac{1}{2}\erfc@{\eta\sqrt{a/2}}+S(a,\eta)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincGammaQ@{a}{z} = \tfrac{1}{2}\erfc@{\eta\sqrt{a/2}}+S(a,\eta)</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, z)/GAMMA(a) = (1)/(2)*erfc(eta*sqrt(a/2))+ S(a , eta)</syntaxhighlight> || <syntaxhighlight lang=mathematica>GammaRegularized[a, z] == Divide[1,2]*Erfc[\[Eta]*Sqrt[a/2]]+ S[a , \[Eta]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.8724483631+.3325384943*I-(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {S = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.8436948582+.7685914925*I-(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {S = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
|-
| [https://dlmf.nist.gov/8.12.E5 8.12.E5] || [[Item:Q2599|<math>\frac{e^{+\pi ia}}{2i\sin@{\pi a}}\normincGammaQ@{-a}{ze^{+\pi i}} = +\tfrac{1}{2}\erfc@{+ i\eta\sqrt{a/2}}-iT(a,\eta)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{e^{+\pi ia}}{2i\sin@{\pi a}}\normincGammaQ@{-a}{ze^{+\pi i}} = +\tfrac{1}{2}\erfc@{+ i\eta\sqrt{a/2}}-iT(a,\eta)</syntaxhighlight> || <math>\realpart@@{-a} > 0</math> || <syntaxhighlight lang=mathematica>(exp(+ Pi*I*a))/(2*I*sin(Pi*a))*GAMMA(- a, z*exp(+ Pi*I))/GAMMA(- a) = +(1)/(2)*erfc(+ I*eta*sqrt(a/2))- I*T(a , eta)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Exp[+ Pi*I*a],2*I*Sin[Pi*a]]*GammaRegularized[- a, z*Exp[+ Pi*I]] == +Divide[1,2]*Erfc[+ I*\[Eta]*Sqrt[a/2]]- I*T[a , \[Eta]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1738836865-.4215091763*I+(-.5000000000+.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5322485765+.1038051776*I+(-.5000000000+.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
|-
| [https://dlmf.nist.gov/8.12.E5 8.12.E5] || [[Item:Q2599|<math>\frac{e^{-\pi ia}}{2i\sin@{\pi a}}\normincGammaQ@{-a}{ze^{-\pi i}} = -\tfrac{1}{2}\erfc@{- i\eta\sqrt{a/2}}-iT(a,\eta)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{e^{-\pi ia}}{2i\sin@{\pi a}}\normincGammaQ@{-a}{ze^{-\pi i}} = -\tfrac{1}{2}\erfc@{- i\eta\sqrt{a/2}}-iT(a,\eta)</syntaxhighlight> || <math>\realpart@@{-a} > 0</math> || <syntaxhighlight lang=mathematica>(exp(- Pi*I*a))/(2*I*sin(Pi*a))*GAMMA(- a, z*exp(- Pi*I))/GAMMA(- a) = -(1)/(2)*erfc(- I*eta*sqrt(a/2))- I*T(a , eta)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Exp[- Pi*I*a],2*I*Sin[Pi*a]]*GammaRegularized[- a, z*Exp[- Pi*I]] == -Divide[1,2]*Erfc[- I*\[Eta]*Sqrt[a/2]]- I*T[a , \[Eta]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.9809290254+.1461521622*I+(-.5000000000+.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.2747967621-.3791621909*I+(-.5000000000+.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
|-
| [https://dlmf.nist.gov/8.12#Ex5 8.12#Ex5] || [[Item:Q2600|<math>\EulerGamma@{a+1}\frac{e^{+\pi ia}}{2\pi i}\incGamma@{-a}{ze^{+\pi i}} = -\tfrac{1}{2}\erfc@{+ i\eta\sqrt{a/2}}+iT(a,\eta)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{a+1}\frac{e^{+\pi ia}}{2\pi i}\incGamma@{-a}{ze^{+\pi i}} = -\tfrac{1}{2}\erfc@{+ i\eta\sqrt{a/2}}+iT(a,\eta)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(a + 1)*(exp(+ Pi*I*a))/(2*Pi*I)*GAMMA(- a, z*exp(+ Pi*I)) = -(1)/(2)*erfc(+ I*eta*sqrt(a/2))+ I*T(a , eta)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a + 1]*Divide[Exp[+ Pi*I*a],2*Pi*I]*Gamma[- a, z*Exp[+ Pi*I]] == -Divide[1,2]*Erfc[+ I*\[Eta]*Sqrt[a/2]]+ I*T[a , \[Eta]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.1738836865+.4215091762*I+(.5000000000-.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .5322485766-.1038051776*I+(.5000000000-.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
|-
| [https://dlmf.nist.gov/8.12#Ex5 8.12#Ex5] || [[Item:Q2600|<math>\EulerGamma@{a+1}\frac{e^{-\pi ia}}{2\pi i}\incGamma@{-a}{ze^{-\pi i}} = +\tfrac{1}{2}\erfc@{- i\eta\sqrt{a/2}}+iT(a,\eta)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{a+1}\frac{e^{-\pi ia}}{2\pi i}\incGamma@{-a}{ze^{-\pi i}} = +\tfrac{1}{2}\erfc@{- i\eta\sqrt{a/2}}+iT(a,\eta)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(a + 1)*(exp(- Pi*I*a))/(2*Pi*I)*GAMMA(- a, z*exp(- Pi*I)) = +(1)/(2)*erfc(- I*eta*sqrt(a/2))+ I*T(a , eta)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a + 1]*Divide[Exp[- Pi*I*a],2*Pi*I]*Gamma[- a, z*Exp[- Pi*I]] == +Divide[1,2]*Erfc[- I*\[Eta]*Sqrt[a/2]]+ I*T[a , \[Eta]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9809290254-.1461521621*I+(.5000000000-.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2747967620+.3791621907*I+(.5000000000-.8660254040*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
|-
| [https://dlmf.nist.gov/8.12.E6 8.12.E6] || [[Item:Q2601|<math>z^{-a}\scincgamma@{-a}{-z} = \cos@{\pi a}-2\sin@{\pi a}\left(\frac{e^{\frac{1}{2}a\eta^{2}}}{\sqrt{\pi}}\DawsonsintF@{\eta\sqrt{a/2}}+T(a,\eta)\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z^{-a}\scincgamma@{-a}{-z} = \cos@{\pi a}-2\sin@{\pi a}\left(\frac{e^{\frac{1}{2}a\eta^{2}}}{\sqrt{\pi}}\DawsonsintF@{\eta\sqrt{a/2}}+T(a,\eta)\right)</syntaxhighlight> || <math>\realpart@@{-a} > 0</math> || <syntaxhighlight lang=mathematica>(z)^(- a)* (- z)^(-(- a))*(GAMMA(- a)-GAMMA(- a, - z))/GAMMA(- a) = cos(Pi*a)- 2*sin(Pi*a)*((exp((1)/(2)*a*(eta)^(2)))/(sqrt(Pi))*dawson(eta*sqrt(a/2))+ T(a , eta))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Failure || Missing Macro Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .2923043261+1.961858052*I+(1.732050808+1.000000000*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.7583243808+.5495935246*I+(1.732050808+1.000000000*I)*(-1.5, .8660254040+.5000000000*I)
Test Values: {T = 1/2*3^(1/2)+1/2*I, a = -1.5, eta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || -
|-
| [https://dlmf.nist.gov/8.12#Ex6 8.12#Ex6] || [[Item:Q2604|<math>c_{0}(\eta) = \frac{1}{\mu}-\frac{1}{\eta}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c_{0}(\eta) = \frac{1}{\mu}-\frac{1}{\eta}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>c[0](eta) = (1)/(mu)-(1)/(eta)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[c, 0][\[Eta]] == Divide[1,\[Mu]]-Divide[1,\[Eta]]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex7 8.12#Ex7] || [[Item:Q2605|<math>c_{1}(\eta) = \frac{1}{\eta^{3}}-\frac{1}{\mu^{3}}-\frac{1}{\mu^{2}}-\frac{1}{12\mu}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c_{1}(\eta) = \frac{1}{\eta^{3}}-\frac{1}{\mu^{3}}-\frac{1}{\mu^{2}}-\frac{1}{12\mu}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>c[1](eta) = (1)/((eta)^(3))-(1)/((mu)^(3))-(1)/((mu)^(2))-(1)/(12*mu)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[c, 1][\[Eta]] == Divide[1,\[Eta]^(3)]-Divide[1,\[Mu]^(3)]-Divide[1,\[Mu]^(2)]-Divide[1,12*\[Mu]]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12.E10 8.12.E10] || [[Item:Q2606|<math>c_{k}(\eta) = \frac{1}{\eta}\deriv{}{\eta}c_{k-1}(\eta)+(-1)^{k}\frac{g_{k}}{\mu}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c_{k}(\eta) = \frac{1}{\eta}\deriv{}{\eta}c_{k-1}(\eta)+(-1)^{k}\frac{g_{k}}{\mu}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>c[k](eta) = (1)/(eta)*diff(c[k - 1](eta), eta)+(- 1)^(k)*(g[k])/(mu)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[c, k][\[Eta]] == Divide[1,\[Eta]]*D[Subscript[c, k - 1][\[Eta]], \[Eta]]+(- 1)^(k)*Divide[Subscript[g, k],\[Mu]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .5000000004+.8660254040*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, c[k] = 1/2*3^(1/2)+1/2*I, c[k-1] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 1, k = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.500000000+.8660254040*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, c[k] = 1/2*3^(1/2)+1/2*I, c[k-1] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 2, k = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.5000000000000001, 0.8660254037844386]
Test Values: {Rule[k, 1], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, Plus[-1, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.5, 0.8660254037844386]
Test Values: {Rule[k, 2], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, Plus[-1, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.12.E11 8.12.E11] || [[Item:Q2607|<math>c_{k}(\eta) = \sum_{n=0}^{\infty}d_{k,n}\eta^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c_{k}(\eta) = \sum_{n=0}^{\infty}d_{k,n}\eta^{n}</syntaxhighlight> || <math>|\eta| < 2\sqrt{\pi}</math> || <syntaxhighlight lang=mathematica>c[k](eta) = sum(d[k , n]*(eta)^(n), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[c, k][\[Eta]] == Sum[Subscript[d, k , n]*\[Eta]^(n), {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex8 8.12#Ex8] || [[Item:Q2608|<math>d_{0,n} = (n+2)\alpha_{n+2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>d_{0,n} = (n+2)\alpha_{n+2}</syntaxhighlight> || <math>n \geq 1</math> || <syntaxhighlight lang=mathematica>d[0 , n] = (n + 2)*alpha[n + 2]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[d, 0 , n] == (n + 2)*Subscript[\[Alpha], n + 2]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex9 8.12#Ex9] || [[Item:Q2609|<math>d_{k,n} = (-1)^{k}g_{k}d_{0,n}+(n+2)d_{k-1,n+2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>d_{k,n} = (-1)^{k}g_{k}d_{0,n}+(n+2)d_{k-1,n+2}</syntaxhighlight> || <math>n \geq 0, k \geq 1</math> || <syntaxhighlight lang=mathematica>d[k , n] = (- 1)^(k)* g[k]*d[0 , n]+(n + 2)*d[k - 1 , n + 2]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[d, k , n] == (- 1)^(k)* Subscript[g, k]*Subscript[d, 0 , n]+(n + 2)*Subscript[d, k - 1 , n + 2]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12.E13 8.12.E13] || [[Item:Q2610|<math>\lambda-1 = \eta+\tfrac{1}{3}\eta^{2}+\sum_{n=3}^{\infty}\alpha_{n}\eta^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\lambda-1 = \eta+\tfrac{1}{3}\eta^{2}+\sum_{n=3}^{\infty}\alpha_{n}\eta^{n}</syntaxhighlight> || <math>|\eta| < 2\sqrt{\pi}</math> || <syntaxhighlight lang=mathematica>lambda - 1 = eta +(1)/(3)*(eta)^(2)+ sum(alpha[n]*(eta)^(n), n = 3..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Lambda]- 1 == \[Eta]+Divide[1,3]*\[Eta]^(2)+ Sum[Subscript[\[Alpha], n]*\[Eta]^(n), {n, 3, Infinity}, GenerateConditions->None]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex10 8.12#Ex10] || [[Item:Q2611|<math>\alpha_{3} = \tfrac{1}{36}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\alpha_{3} = \tfrac{1}{36}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>alpha[3] = (1)/(36)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Alpha], 3] == Divide[1,36]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex11 8.12#Ex11] || [[Item:Q2612|<math>\alpha_{4} = -\tfrac{1}{270}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\alpha_{4} = -\tfrac{1}{270}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>alpha[4] = -(1)/(270)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Alpha], 4] == -Divide[1,270]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex12 8.12#Ex12] || [[Item:Q2613|<math>\alpha_{5} = \tfrac{1}{4320}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\alpha_{5} = \tfrac{1}{4320}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>alpha[5] = (1)/(4320)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Alpha], 5] == Divide[1,4320]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex13 8.12#Ex13] || [[Item:Q2614|<math>\alpha_{6} = \tfrac{1}{17010}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\alpha_{6} = \tfrac{1}{17010}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>alpha[6] = (1)/(17010)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Alpha], 6] == Divide[1,17010]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex14 8.12#Ex14] || [[Item:Q2615|<math>\alpha_{7} = -\tfrac{139}{54\;43200}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\alpha_{7} = -\tfrac{139}{54\;43200}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>alpha[7] = -(139)/(5443200)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Alpha], 7] == -Divide[139,5443200]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex15 8.12#Ex15] || [[Item:Q2616|<math>\alpha_{8} = \tfrac{1}{2\;04120}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\alpha_{8} = \tfrac{1}{2\;04120}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>alpha[8] = (1)/(204120)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[\[Alpha], 8] == Divide[1,204120]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex16 8.12#Ex16] || [[Item:Q2619|<math>c_{0}(0) = -\tfrac{1}{3}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c_{0}(0) = -\tfrac{1}{3}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>c[0](0) = -(1)/(3)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[c, 0][0] == -Divide[1,3]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex17 8.12#Ex17] || [[Item:Q2620|<math>c_{1}(0) = -\tfrac{1}{540}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c_{1}(0) = -\tfrac{1}{540}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>c[1](0) = -(1)/(540)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[c, 1][0] == -Divide[1,540]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex18 8.12#Ex18] || [[Item:Q2621|<math>c_{2}(0) = \tfrac{25}{6048}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c_{2}(0) = \tfrac{25}{6048}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>c[2](0) = (25)/(6048)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[c, 2][0] == Divide[25,6048]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex19 8.12#Ex19] || [[Item:Q2622|<math>c_{3}(0) = \tfrac{101}{1\;55520}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c_{3}(0) = \tfrac{101}{1\;55520}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>c[3](0) = (101)/(155520)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[c, 3][0] == Divide[101,155520]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex20 8.12#Ex20] || [[Item:Q2623|<math>c_{4}(0) = -\tfrac{31\;84811}{36951\;55200}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c_{4}(0) = -\tfrac{31\;84811}{36951\;55200}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>c[4](0) = -(3184811)/(3695155200)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[c, 4][0] == -Divide[3184811,3695155200]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12#Ex21 8.12#Ex21] || [[Item:Q2624|<math>c_{5}(0) = -\tfrac{27\;45493}{81517\;36320}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c_{5}(0) = -\tfrac{27\;45493}{81517\;36320}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>c[5](0) = -(2745493)/(8151736320)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[c, 5][0] == -Divide[2745493,8151736320]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.12.E21 8.12.E21] || [[Item:Q2631|<math>\normincGammaQ@{a}{x} = q</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincGammaQ@{a}{x} = q</syntaxhighlight> || <math>\realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a, x)/GAMMA(a) = q</syntaxhighlight> || <syntaxhighlight lang=mathematica>GammaRegularized[a, x] == q</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.8512854784-.5000000000*I
Test Values: {a = -1.5, q = 1/2*3^(1/2)+1/2*I, x = 1.5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5487148961-.5000000000*I
Test Values: {a = -1.5, q = 1/2*3^(1/2)+1/2*I, x = .5}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.8512854781447857, -0.49999999999999994]
Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.5487148959215247, -0.49999999999999994]
Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.13.E1 8.13.E1] || [[Item:Q2633|<math>x_{-}(a) < \ln@@{|a|}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x_{-}(a) < \ln@@{|a|}</syntaxhighlight> || <math>-1 < a, a < 0</math> || <syntaxhighlight lang=mathematica>x[-](a) < ln(abs(a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[x, -][a] < Log[Abs[a]]</syntaxhighlight> || Error || Failure || - || Error
|-
| [https://dlmf.nist.gov/8.14.E1 8.14.E1] || [[Item:Q2634|<math>\int_{0}^{\infty}e^{-ax}\frac{\incgamma@{b}{x}}{\EulerGamma@{b}}\diff{x} = \frac{(1+a)^{-b}}{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-ax}\frac{\incgamma@{b}{x}}{\EulerGamma@{b}}\diff{x} = \frac{(1+a)^{-b}}{a}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > -1, \realpart@@{b} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- a*x)*(GAMMA(b)-GAMMA(b, x))/(GAMMA(b)), x = 0..infinity) = ((1 + a)^(- b))/(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- a*x]*Divide[Gamma[b, 0, x],Gamma[b]], {x, 0, Infinity}, GenerateConditions->None] == Divide[(1 + a)^(- b),a]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.14.E2 8.14.E2] || [[Item:Q2635|<math>\int_{0}^{\infty}e^{-ax}\incGamma@{b}{x}\diff{x} = \EulerGamma@{b}\frac{1-(1+a)^{-b}}{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-ax}\incGamma@{b}{x}\diff{x} = \EulerGamma@{b}\frac{1-(1+a)^{-b}}{a}</syntaxhighlight> || <math>\realpart@@{a} > -1, \realpart@@{b} > -1, \realpart@@{b} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- a*x)*GAMMA(b, x), x = 0..infinity) = GAMMA(b)*(1 -(1 + a)^(- b))/(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- a*x]*Gamma[b, x], {x, 0, Infinity}, GenerateConditions->None] == Gamma[b]*Divide[1 -(1 + a)^(- b),a]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 12] || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.14.E3 8.14.E3] || [[Item:Q2636|<math>\int_{0}^{\infty}x^{a-1}\incgamma@{b}{x}\diff{x} = -\frac{\EulerGamma@{a+b}}{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}x^{a-1}\incgamma@{b}{x}\diff{x} = -\frac{\EulerGamma@{a+b}}{a}</syntaxhighlight> || <math>\realpart@@{a} < 0, \realpart@{a+b} > 0, \realpart@@{a+b} > 0, \realpart@@{b} > 0</math> || <syntaxhighlight lang=mathematica>int((x)^(a - 1)* GAMMA(b)-GAMMA(b, x), x = 0..infinity) = -(GAMMA(a + b))/(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(x)^(a - 1)* Gamma[b, 0, x], {x, 0, Infinity}, GenerateConditions->None] == -Divide[Gamma[a + b],a]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)
Test Values: {a = -1.5, b = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)
Test Values: {a = -.5, b = 1.5}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skip - No test values generated
|-
| [https://dlmf.nist.gov/8.14.E4 8.14.E4] || [[Item:Q2637|<math>\int_{0}^{\infty}x^{a-1}\incGamma@{b}{x}\diff{x} = \frac{\EulerGamma@{a+b}}{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}x^{a-1}\incGamma@{b}{x}\diff{x} = \frac{\EulerGamma@{a+b}}{a}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@{a+b} > 0, \realpart@@{a+b} > 0</math> || <syntaxhighlight lang=mathematica>int((x)^(a - 1)* GAMMA(b, x), x = 0..infinity) = (GAMMA(a + b))/(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(x)^(a - 1)* Gamma[b, x], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[a + b],a]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 12]
|-
| [https://dlmf.nist.gov/8.14.E5 8.14.E5] || [[Item:Q2638|<math>\int_{0}^{\infty}x^{a-1}e^{-sx}\incgamma@{b}{x}\diff{x} = \frac{\EulerGamma@{a+b}}{b(1+s)^{a+b}}\*\hyperF@{1}{a+b}{1+b}{1/(1+s)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}x^{a-1}e^{-sx}\incgamma@{b}{x}\diff{x} = \frac{\EulerGamma@{a+b}}{b(1+s)^{a+b}}\*\hyperF@{1}{a+b}{1+b}{1/(1+s)}</syntaxhighlight> || <math>\realpart@@{s} > 0, \realpart@{a+b} > 0, \realpart@@{a+b} > 0, \realpart@@{b} > 0</math> || <syntaxhighlight lang=mathematica>int((x)^(a - 1)* exp(- s*x)*GAMMA(b)-GAMMA(b, x), x = 0..infinity) = (GAMMA(a + b))/(b*(1 + s)^(a + b))* hypergeom([1, a + b], [1 + b], 1/(1 + s))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(x)^(a - 1)* Exp[- s*x]*Gamma[b, 0, x], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[a + b],b*(1 + s)^(a + b)]* Hypergeometric2F1[1, a + b, 1 + b, 1/(1 + s)]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [36 / 36]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)
Test Values: {a = -1.5, b = 2, s = 1.5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)
Test Values: {a = -1.5, b = 2, s = .5}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.14.E6 8.14.E6] || [[Item:Q2639|<math>\int_{0}^{\infty}x^{a-1}e^{-sx}\incGamma@{b}{x}\diff{x} = \frac{\EulerGamma@{a+b}}{a(1+s)^{a+b}}\*\hyperF@{1}{a+b}{1+a}{s/(1+s)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}x^{a-1}e^{-sx}\incGamma@{b}{x}\diff{x} = \frac{\EulerGamma@{a+b}}{a(1+s)^{a+b}}\*\hyperF@{1}{a+b}{1+a}{s/(1+s)}</syntaxhighlight> || <math>\realpart@@{s} > -1, \realpart@{a+b} > 0, \realpart@@{a} > 0, \realpart@@{a+b} > 0</math> || <syntaxhighlight lang=mathematica>int((x)^(a - 1)* exp(- s*x)*GAMMA(b, x), x = 0..infinity) = (GAMMA(a + b))/(a*(1 + s)^(a + b))* hypergeom([1, a + b], [1 + a], s/(1 + s))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(x)^(a - 1)* Exp[- s*x]*Gamma[b, x], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[a + b],a*(1 + s)^(a + b)]* Hypergeometric2F1[1, a + b, 1 + a, s/(1 + s)]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.15.E1 8.15.E1] || [[Item:Q2640|<math>\incgamma@{a}{\lambda x} = \lambda^{a}\sum_{k=0}^{\infty}\incgamma@{a+k}{x}\frac{(1-\lambda)^{k}}{k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incgamma@{a}{\lambda x} = \lambda^{a}\sum_{k=0}^{\infty}\incgamma@{a+k}{x}\frac{(1-\lambda)^{k}}{k!}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{a+k} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(a)-GAMMA(a, lambda*x) = (lambda)^(a)* sum(GAMMA(a + k)-GAMMA(a + k, x)*((1 - lambda)^(k))/(factorial(k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[a, 0, \[Lambda]*x] == \[Lambda]^(a)* Sum[Gamma[a + k, 0, x]*Divide[(1 - \[Lambda])^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [90 / 90]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {a = 1.5, lambda = 1/2*3^(1/2)+1/2*I, x = 1.5}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {a = 1.5, lambda = 1/2*3^(1/2)+1/2*I, x = .5}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.17.E1 8.17.E1] || [[Item:Q2641|<math>\incBeta{x}@{a}{b} = \int_{0}^{x}t^{a-1}(1-t)^{b-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incBeta{x}@{a}{b} = \int_{0}^{x}t^{a-1}(1-t)^{b-1}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(t^(a-1)*(1-t)^(b-1), t = 0 .. x) = int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Beta[x, a, b] == Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, x}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 108]
|-
| [https://dlmf.nist.gov/8.17.E2 8.17.E2] || [[Item:Q2642|<math>\normincBetaI{x}@{a}{b} = \incBeta{x}@{a}{b}/\EulerBeta@{a}{b}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincBetaI{x}@{a}{b} = \incBeta{x}@{a}{b}/\EulerBeta@{a}{b}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{a+b} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BetaRegularized[x, a, b] == Beta[x, a, b]/Beta[a, b]</syntaxhighlight> || Missing Macro Error || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [45 / 108]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.17.E3 8.17.E3] || [[Item:Q2643|<math>\EulerBeta@{a}{b} = \frac{\EulerGamma@{a}\EulerGamma@{b}}{\EulerGamma@{a+b}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerBeta@{a}{b} = \frac{\EulerGamma@{a}\EulerGamma@{b}}{\EulerGamma@{a+b}}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{a+b} > 0</math> || <syntaxhighlight lang=mathematica>Beta(a, b) = (GAMMA(a)*GAMMA(b))/(GAMMA(a + b))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Beta[a, b] == Divide[Gamma[a]*Gamma[b],Gamma[a + b]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 9] || Successful [Tested: 9]
|-
| [https://dlmf.nist.gov/8.17.E4 8.17.E4] || [[Item:Q2644|<math>\normincBetaI{x}@{a}{b} = 1-\normincBetaI{1-x}@{b}{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincBetaI{x}@{a}{b} = 1-\normincBetaI{1-x}@{b}{a}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{a+b} > 0, \realpart@@{b+b} > 0, \realpart@@{a+a} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BetaRegularized[x, a, b] == 1 - BetaRegularized[1 - x, b, a]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [39 / 108]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.17.E5 8.17.E5] || [[Item:Q2645|<math>\normincBetaI{x}@{m}{n-m+1} = \sum_{j=m}^{n}\binom{n}{j}x^{j}(1-x)^{n-j}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincBetaI{x}@{m}{n-m+1} = \sum_{j=m}^{n}\binom{n}{j}x^{j}(1-x)^{n-j}</syntaxhighlight> || <math>0 \leq x, x < 1, \realpart@@{m} > 0, \realpart@@{n-m+1} > 0, \realpart@@{m+b} > 0, \realpart@@{a+n-m+1} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BetaRegularized[x, m, n - m + 1] == Sum[Binomial[n,j]*(x)^(j)*(1 - x)^(n - j), {j, m, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 9]
|-
| [https://dlmf.nist.gov/8.17.E6 8.17.E6] || [[Item:Q2646|<math>\normincBetaI{x}@{a}{a} = \tfrac{1}{2}\normincBetaI{4x(1-x)}@{a}{\tfrac{1}{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincBetaI{x}@{a}{a} = \tfrac{1}{2}\normincBetaI{4x(1-x)}@{a}{\tfrac{1}{2}}</syntaxhighlight> || <math>0 \leq x, x \leq \frac{1}{2}, \realpart@@{a} > 0, \realpart@@{\tfrac{1}{2}} > 0, \realpart@@{a+b} > 0, \realpart@@{a+a} > 0, \realpart@@{a+\tfrac{1}{2}} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BetaRegularized[x, a, a] == Divide[1,2]*BetaRegularized[4*x*(1 - x), a, Divide[1,2]]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[x, 0.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -0.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.17.E7 8.17.E7] || [[Item:Q2647|<math>\incBeta{x}@{a}{b} = \frac{x^{a}}{a}\hyperF@{a}{1-b}{a+1}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incBeta{x}@{a}{b} = \frac{x^{a}}{a}\hyperF@{a}{1-b}{a+1}{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(t^(a-1)*(1-t)^(b-1), t = 0 .. x) = ((x)^(a))/(a)*hypergeom([a, 1 - b], [a + 1], x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Beta[x, a, b] == Divide[(x)^(a),a]*Hypergeometric2F1[a, 1 - b, a + 1, x]</syntaxhighlight> || Failure || Successful || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 108]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -2], Rule[b, -1.5], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -2], Rule[b, -1.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.17.E8 8.17.E8] || [[Item:Q2648|<math>\incBeta{x}@{a}{b} = \frac{x^{a}(1-x)^{b}}{a}\hyperF@{a+b}{1}{a+1}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incBeta{x}@{a}{b} = \frac{x^{a}(1-x)^{b}}{a}\hyperF@{a+b}{1}{a+1}{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(t^(a-1)*(1-t)^(b-1), t = 0 .. x) = ((x)^(a)*(1 - x)^(b))/(a)*hypergeom([a + b, 1], [a + 1], x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Beta[x, a, b] == Divide[(x)^(a)*(1 - x)^(b),a]*Hypergeometric2F1[a + b, 1, a + 1, x]</syntaxhighlight> || Failure || Successful || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [15 / 108]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -2], Rule[b, -1.5], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -2], Rule[b, -1.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.17.E9 8.17.E9] || [[Item:Q2649|<math>\incBeta{x}@{a}{b} = \frac{x^{a}(1-x)^{b-1}}{a}\hyperF@@{1}{1-b}{a+1}{\frac{x}{x-1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\incBeta{x}@{a}{b} = \frac{x^{a}(1-x)^{b-1}}{a}\hyperF@@{1}{1-b}{a+1}{\frac{x}{x-1}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(t^(a-1)*(1-t)^(b-1), t = 0 .. x) = ((x)^(a)*(1 - x)^(b - 1))/(a)*hypergeom([1, 1 - b], [a + 1], (x)/(x - 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Beta[x, a, b] == Divide[(x)^(a)*(1 - x)^(b - 1),a]*Hypergeometric2F1[1, 1 - b, a + 1, Divide[x,x - 1]]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [46 / 108]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[4.9960036108132044*^-15, -27.48893571891068]
Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[3.191891195797325*^-15, -27.48893571891068]
Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[x, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.17.E10 8.17.E10] || [[Item:Q2650|<math>\normincBetaI{x}@{a}{b} = \frac{x^{a}(1-x)^{b}}{2\pi i}\int_{c-i\infty}^{c+i\infty}s^{-a}(1-s)^{-b}\frac{\diff{s}}{s-x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincBetaI{x}@{a}{b} = \frac{x^{a}(1-x)^{b}}{2\pi i}\int_{c-i\infty}^{c+i\infty}s^{-a}(1-s)^{-b}\frac{\diff{s}}{s-x}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{a+b} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BetaRegularized[x, a, b] == Divide[(x)^(a)*(1 - x)^(b),2*Pi*I]*Integrate[(s)^(- a)*(1 - s)^(- b)*Divide[1,s - x], {s, c - I*Infinity, c + I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.17.E12 8.17.E12] || [[Item:Q2653|<math>\normincBetaI{x}@{a}{b} = x\normincBetaI{x}@{a-1}{b}+x^{\prime}\normincBetaI{x}@{a}{b-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincBetaI{x}@{a}{b} = x\normincBetaI{x}@{a-1}{b}+x^{\prime}\normincBetaI{x}@{a}{b-1}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{a-1} > 0, \realpart@@{b} > 0, \realpart@@{b-1} > 0, \realpart@@{a+b} > 0, \realpart@@{a-1+b} > 0, \realpart@@{a+b-1} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BetaRegularized[x, a, b] == x*BetaRegularized[x, a - 1, b]+(1 - x)*BetaRegularized[x, a, b - 1]</syntaxhighlight> || Missing Macro Error || Failure || - || Error
|-
| [https://dlmf.nist.gov/8.17.E13 8.17.E13] || [[Item:Q2654|<math>(a+b)\normincBetaI{x}@{a}{b} = a\normincBetaI{x}@{a+1}{b}+b\normincBetaI{x}@{a}{b+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(a+b)\normincBetaI{x}@{a}{b} = a\normincBetaI{x}@{a+1}{b}+b\normincBetaI{x}@{a}{b+1}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{a+1} > 0, \realpart@@{b} > 0, \realpart@@{b+1} > 0, \realpart@@{a+b} > 0, \realpart@@{a+1+b} > 0, \realpart@@{a+b+1} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(a + b)*BetaRegularized[x, a, b] == a*BetaRegularized[x, a + 1, b]+ b*BetaRegularized[x, a, b + 1]</syntaxhighlight> || Missing Macro Error || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [33 / 108]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.17.E14 8.17.E14] || [[Item:Q2655|<math>(a+bx)\normincBetaI{x}@{a}{b} = xb\normincBetaI{x}@{a-1}{b+1}+a\normincBetaI{x}@{a+1}{b}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(a+bx)\normincBetaI{x}@{a}{b} = xb\normincBetaI{x}@{a-1}{b+1}+a\normincBetaI{x}@{a+1}{b}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{a-1} > 0, \realpart@@{a+1} > 0, \realpart@@{b} > 0, \realpart@@{b+1} > 0, \realpart@@{a+b} > 0, \realpart@@{a-1+b} > 0, \realpart@@{a+1+b} > 0, \realpart@@{a+b+1} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(a + b*x)*BetaRegularized[x, a, b] == x*b*BetaRegularized[x, a - 1, b + 1]+ a*BetaRegularized[x, a + 1, b]</syntaxhighlight> || Missing Macro Error || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [36 / 108]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.17.E16 8.17.E16] || [[Item:Q2657|<math>a\normincBetaI{x}@{a+1}{b} = (a+cx)\normincBetaI{x}@{a}{b}-cx\normincBetaI{x}@{a-1}{b}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>a\normincBetaI{x}@{a+1}{b} = (a+cx)\normincBetaI{x}@{a}{b}-cx\normincBetaI{x}@{a-1}{b}</syntaxhighlight> || <math>\realpart@@{a+1} > 0, \realpart@@{a} > 0, \realpart@@{a-1} > 0, \realpart@@{b} > 0, \realpart@@{a+1+b} > 0, \realpart@@{a+b} > 0, \realpart@@{a-1+b} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>a*BetaRegularized[x, a + 1, b] == (a + c*x)*BetaRegularized[x, a, b]- c*x*BetaRegularized[x, a - 1, b]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [246 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.17.E18 8.17.E18] || [[Item:Q2659|<math>\normincBetaI{x}@{a}{b} = \normincBetaI{x}@{a+1}{b-1}+\frac{x^{a}(x^{\prime})^{b-1}}{a\EulerBeta@{a}{b}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincBetaI{x}@{a}{b} = \normincBetaI{x}@{a+1}{b-1}+\frac{x^{a}(x^{\prime})^{b-1}}{a\EulerBeta@{a}{b}}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{a+b} > 0, \realpart@@{a+1} > 0, \realpart@@{b-1} > 0, \realpart@@{a+1+b} > 0, \realpart@@{a+b-1} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BetaRegularized[x, a, b] == BetaRegularized[x, a + 1, b - 1]+Divide[(x)^(a)*((1 - x))^(b - 1),a*Beta[a, b]]</syntaxhighlight> || Missing Macro Error || Failure || - || Error
|-
| [https://dlmf.nist.gov/8.17.E19 8.17.E19] || [[Item:Q2660|<math>\normincBetaI{x}@{a}{b} = \normincBetaI{x}@{a-1}{b+1}-\frac{x^{a-1}(x^{\prime})^{b}}{b\EulerBeta@{a}{b}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincBetaI{x}@{a}{b} = \normincBetaI{x}@{a-1}{b+1}-\frac{x^{a-1}(x^{\prime})^{b}}{b\EulerBeta@{a}{b}}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{a+b} > 0, \realpart@@{a-1} > 0, \realpart@@{b+1} > 0, \realpart@@{a-1+b} > 0, \realpart@@{a+b+1} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BetaRegularized[x, a, b] == BetaRegularized[x, a - 1, b + 1]-Divide[(x)^(a - 1)*((1 - x))^(b),b*Beta[a, b]]</syntaxhighlight> || Missing Macro Error || Failure || - || Error
|-
| [https://dlmf.nist.gov/8.17.E20 8.17.E20] || [[Item:Q2661|<math>\normincBetaI{x}@{a}{b} = \normincBetaI{x}@{a+1}{b}+\frac{x^{a}(x^{\prime})^{b}}{a\EulerBeta@{a}{b}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincBetaI{x}@{a}{b} = \normincBetaI{x}@{a+1}{b}+\frac{x^{a}(x^{\prime})^{b}}{a\EulerBeta@{a}{b}}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{a+b} > 0, \realpart@@{a+1} > 0, \realpart@@{a+1+b} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BetaRegularized[x, a, b] == BetaRegularized[x, a + 1, b]+Divide[(x)^(a)*((1 - x))^(b),a*Beta[a, b]]</syntaxhighlight> || Missing Macro Error || Failure || - || Error
|-
| [https://dlmf.nist.gov/8.17.E21 8.17.E21] || [[Item:Q2662|<math>\normincBetaI{x}@{a}{b} = \normincBetaI{x}@{a}{b+1}-\frac{x^{a}(x^{\prime})^{b}}{b\EulerBeta@{a}{b}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincBetaI{x}@{a}{b} = \normincBetaI{x}@{a}{b+1}-\frac{x^{a}(x^{\prime})^{b}}{b\EulerBeta@{a}{b}}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{a+b} > 0, \realpart@@{b+1} > 0, \realpart@@{a+b+1} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BetaRegularized[x, a, b] == BetaRegularized[x, a, b + 1]-Divide[(x)^(a)*((1 - x))^(b),b*Beta[a, b]]</syntaxhighlight> || Missing Macro Error || Failure || - || Error
|-
| [https://dlmf.nist.gov/8.17#Ex3 8.17#Ex3] || [[Item:Q2664|<math>d_{2m} = \frac{m(b-m)x}{(a+2m-1)(a+2m)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>d_{2m} = \frac{m(b-m)x}{(a+2m-1)(a+2m)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>d[2*m] = (m*(b - m)*x)/((a + 2*m - 1)*(a + 2*m))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[d, 2*m] == Divide[m*(b - m)*x,(a + 2*m - 1)*(a + 2*m)]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.17#Ex4 8.17#Ex4] || [[Item:Q2665|<math>d_{2m+1} = -\frac{(a+m)(a+b+m)x}{(a+2m)(a+2m+1)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>d_{2m+1} = -\frac{(a+m)(a+b+m)x}{(a+2m)(a+2m+1)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>d[2*m + 1] = -((a + m)*(a + b + m)*x)/((a + 2*m)*(a + 2*m + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[d, 2*m + 1] == -Divide[(a + m)*(a + b + m)*x,(a + 2*m)*(a + 2*m + 1)]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.17.E24 8.17.E24] || [[Item:Q2666|<math>\normincBetaI{x}@{m}{n} = (1-x)^{n}\sum_{j=m}^{\infty}\binom{n+j-1}{j}x^{j}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincBetaI{x}@{m}{n} = (1-x)^{n}\sum_{j=m}^{\infty}\binom{n+j-1}{j}x^{j}</syntaxhighlight> || <math>0 \leq x, x < 1, \realpart@@{m} > 0, \realpart@@{n} > 0, \realpart@@{m+b} > 0, \realpart@@{a+n} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BetaRegularized[x, m, n] == (1 - x)^(n)* Sum[Binomial[n + j - 1,j]*(x)^(j), {j, m, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 9]
|-
| [https://dlmf.nist.gov/8.18.E2 8.18.E2] || [[Item:Q2668|<math>\xi = -\ln@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\xi = -\ln@@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>xi = - ln(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Xi] == - Log[x]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.271490512+.5000000000*I
Test Values: {x = 1.5, xi = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.945348919e-1+.8660254040*I
Test Values: {x = 1.5, xi = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.271490511892603, 0.49999999999999994]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.0945348918918354, 0.8660254037844387]
Test Values: {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.18.E4 8.18.E4] || [[Item:Q2670|<math>aF_{k+1} = (k+b-a\xi)F_{k}+k\xi F_{k-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>aF_{k+1} = (k+b-a\xi)F_{k}+k\xi F_{k-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>a*F[k + 1] = (k + b - a*xi)*F[k]+ k*xi*F[k - 1]</syntaxhighlight> || <syntaxhighlight lang=mathematica>a*Subscript[F, k + 1] == (k + b - a*\[Xi])*Subscript[F, k]+ k*\[Xi]*Subscript[F, k - 1]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.18#Ex1 8.18#Ex1] || [[Item:Q2671|<math>F_{0} = a^{-b}\normincGammaQ@{b}{a\xi}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>F_{0} = a^{-b}\normincGammaQ@{b}{a\xi}</syntaxhighlight> || <math>\realpart@@{b} > 0</math> || <syntaxhighlight lang=mathematica>F[0] = (a)^(- b)* GAMMA(b, a*xi)/GAMMA(b)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[F, 0] == (a)^(- b)* GammaRegularized[b, a*\[Xi]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.253924788+1.407498490*I
Test Values: {a = -1.5, b = -1.5, xi = 1/2*3^(1/2)+1/2*I, F[0] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.1121006157+1.773523894*I
Test Values: {a = -1.5, b = -1.5, xi = 1/2*3^(1/2)+1/2*I, F[0] = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.2539247882576399, 1.4074984905445393]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.11210061552679867, 1.7735238943289782]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 0], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.18#Ex2 8.18#Ex2] || [[Item:Q2672|<math>F_{1} = \frac{b-a\xi}{a}F_{0}+\frac{\xi^{b}e^{-a\xi}}{a\EulerGamma@{b}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>F_{1} = \frac{b-a\xi}{a}F_{0}+\frac{\xi^{b}e^{-a\xi}}{a\EulerGamma@{b}}</syntaxhighlight> || <math>\realpart@@{b} > 0</math> || <syntaxhighlight lang=mathematica>F[1] = (b - a*xi)/(a)*F[0]+((xi)^(b)* exp(- a*xi))/(a*GAMMA(b))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[F, 1] == Divide[b - a*\[Xi],a]*Subscript[F, 0]+Divide[\[Xi]^(b)* Exp[- a*\[Xi]],a*Gamma[b]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.329643864+4.621882749*I
Test Values: {a = -1.5, b = 1.5, xi = 1/2*3^(1/2)+1/2*I, F[0] = 1/2*3^(1/2)+1/2*I, F[1] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .9636184598+4.987908153*I
Test Values: {a = -1.5, b = 1.5, xi = 1/2*3^(1/2)+1/2*I, F[0] = 1/2*3^(1/2)+1/2*I, F[1] = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[2.32964386182885, 4.621882746395113]
Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.9636184580444114, 4.9879081501795515]
Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.18.E6 8.18.E6] || [[Item:Q2673|<math>\left(\frac{1-e^{-t}}{t}\right)^{b-1} = \sum_{k=0}^{\infty}d_{k}(t-\xi)^{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\frac{1-e^{-t}}{t}\right)^{b-1} = \sum_{k=0}^{\infty}d_{k}(t-\xi)^{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>((1 - exp(- t))/(t))^(b - 1) = sum(d[k]*(t - xi)^(k), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Divide[1 - Exp[- t],t])^(b - 1) == Sum[Subscript[d, k]*(t - \[Xi])^(k), {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.18#Ex3 8.18#Ex3] || [[Item:Q2674|<math>d_{0} = \left(\frac{1-x}{\xi}\right)^{b-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>d_{0} = \left(\frac{1-x}{\xi}\right)^{b-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>d[0] = ((1 - x)/(xi))^(b - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[d, 0] == (Divide[1 - x,\[Xi]])^(b - 1)</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.18#Ex4 8.18#Ex4] || [[Item:Q2675|<math>d_{1} = \frac{x\xi+x-1}{(1-x)\xi}(b-1)d_{0}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>d_{1} = \frac{x\xi+x-1}{(1-x)\xi}(b-1)d_{0}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>d[1] = (x*xi + x - 1)/((1 - x)*xi)*(b - 1)*d[0]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[d, 1] == Divide[x*\[Xi]+ x - 1,(1 - x)*\[Xi]]*(b - 1)*Subscript[d, 0]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.18.E8 8.18.E8] || [[Item:Q2676|<math>x_{0} = a/(a+b)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x_{0} = a/(a+b)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>x[0] = a/(a + b)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[x, 0] == a/(a + b)</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.18.E10 8.18.E10] || [[Item:Q2678|<math>-\tfrac{1}{2}\eta^{2} = x_{0}\ln@{\frac{x}{x_{0}}}+(1-x_{0})\ln@{\frac{1-x}{1-x_{0}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\tfrac{1}{2}\eta^{2} = x_{0}\ln@{\frac{x}{x_{0}}}+(1-x_{0})\ln@{\frac{1-x}{1-x_{0}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>-(1)/(2)*(eta)^(2) = x[0]*ln((x)/(x[0]))+(1 - x[0])*ln((1 - x)/(1 - x[0]))</syntaxhighlight> || <syntaxhighlight lang=mathematica>-Divide[1,2]*\[Eta]^(2) == Subscript[x, 0]*Log[Divide[x,Subscript[x, 0]]]+(1 - Subscript[x, 0])*Log[Divide[1 - x,1 - Subscript[x, 0]]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .580000474e-1+.458917392e-1*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, x = 1.5, x[0] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.269862383+1.019641337*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, x = 1.5, x[0] = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.058000047924774145, 0.04589173995258988]
Test Values: {Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[2.2698623824536366, 1.0196413375539057]
Test Values: {Rule[x, 1.5], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 0], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.18.E11 8.18.E11] || [[Item:Q2679|<math>c_{0}(\eta) = \frac{1}{\eta}-\frac{\sqrt{x_{0}(1-x_{0})}}{x-x_{0}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c_{0}(\eta) = \frac{1}{\eta}-\frac{\sqrt{x_{0}(1-x_{0})}}{x-x_{0}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>c[0](eta) = (1)/(eta)-(sqrt(x[0]*(1 - x[0])))/(x - x[0])</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[c, 0][\[Eta]] == Divide[1,\[Eta]]-Divide[Sqrt[Subscript[x, 0]*(1 - Subscript[x, 0])],x - Subscript[x, 0]]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.18.E12 8.18.E12] || [[Item:Q2680|<math>c_{0}(0) = \frac{1-2x_{0}}{3\sqrt{x_{0}(1-x_{0})}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c_{0}(0) = \frac{1-2x_{0}}{3\sqrt{x_{0}(1-x_{0})}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>c[0](0) = (1 - 2*x[0])/(3*sqrt(x[0]*(1 - x[0])))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[c, 0][0] == Divide[1 - 2*Subscript[x, 0],3*Sqrt[Subscript[x, 0]*(1 - Subscript[x, 0])]]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.18.E15 8.18.E15] || [[Item:Q2683|<math>\mu\ln@@{\zeta}-\zeta = \ln@@{x}+\mu\ln@{1-x}+(1+\mu)\ln@{1+\mu}-\mu</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\mu\ln@@{\zeta}-\zeta = \ln@@{x}+\mu\ln@{1-x}+(1+\mu)\ln@{1+\mu}-\mu</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>mu*ln(zeta)- zeta = ln(x)+ mu*ln(1 - x)+(1 + mu)*ln(1 + mu)- mu</syntaxhighlight> || <syntaxhighlight lang=mathematica>\[Mu]*Log[\[Zeta]]- \[Zeta] == Log[x]+ \[Mu]*Log[1 - x]+(1 + \[Mu])*Log[1 + \[Mu]]- \[Mu]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [299 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .405976146-2.738439399*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, x = 1.5, zeta = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .9866033870-1.744115280*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, x = 1.5, zeta = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [299 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.4059761460255107, -2.7384393975724306]
Test Values: {Rule[x, 1.5], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.0560847852373059, 1.7517066341083583]
Test Values: {Rule[x, 1.5], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.18.E16 8.18.E16] || [[Item:Q2684|<math>h_{0}(\zeta,\mu) = \mu\left(\frac{1}{\zeta-\mu}-\frac{(1+\mu)^{-3/2}}{x_{0}-x}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>h_{0}(\zeta,\mu) = \mu\left(\frac{1}{\zeta-\mu}-\frac{(1+\mu)^{-3/2}}{x_{0}-x}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>h[0](zeta , mu) = mu*((1)/(zeta - mu)-((1 + mu)^(- 3/2))/(x[0]- x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[h, 0][\[Zeta], \[Mu]] == \[Mu]*(Divide[1,\[Zeta]- \[Mu]]-Divide[(1 + \[Mu])^(- 3/2),Subscript[x, 0]- x])</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.18.E17 8.18.E17] || [[Item:Q2685|<math>h_{0}(\mu,\mu) = \frac{1}{3}\left(\frac{1-\mu}{\sqrt{1+\mu}}-1\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>h_{0}(\mu,\mu) = \frac{1}{3}\left(\frac{1-\mu}{\sqrt{1+\mu}}-1\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>h[0](mu , mu) = (1)/(3)*((1 - mu)/(sqrt(1 + mu))- 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[h, 0][\[Mu], \[Mu]] == Divide[1,3]*(Divide[1 - \[Mu],Sqrt[1 + \[Mu]]]- 1)</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.18.E18 8.18.E18] || [[Item:Q2686|<math>\normincBetaI{x}@{a}{b} = p</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\normincBetaI{x}@{a}{b} = p</syntaxhighlight> || <math>0 \leq p, p \leq 1, \realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{a+b} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BetaRegularized[x, a, b] == p</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [105 / 108]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[p, 0.5], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[p, 0.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.19.E1 8.19.E1] || [[Item:Q2687|<math>\genexpintE{p}@{z} = z^{p-1}\incGamma@{1-p}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genexpintE{p}@{z} = z^{p-1}\incGamma@{1-p}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Ei(p, z) = (z)^(p - 1)* GAMMA(1 - p, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ExpIntegralE[p, z] == (z)^(p - 1)* Gamma[1 - p, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 70]
|-
| [https://dlmf.nist.gov/8.19.E2 8.19.E2] || [[Item:Q2688|<math>\genexpintE{p}@{z} = z^{p-1}\int_{z}^{\infty}\frac{e^{-t}}{t^{p}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genexpintE{p}@{z} = z^{p-1}\int_{z}^{\infty}\frac{e^{-t}}{t^{p}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Ei(p, z) = (z)^(p - 1)* int((exp(- t))/((t)^(p)), t = z..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ExpIntegralE[p, z] == (z)^(p - 1)* Integrate[Divide[Exp[- t],(t)^(p)], {t, z, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 70]
|-
| [https://dlmf.nist.gov/8.19.E3 8.19.E3] || [[Item:Q2689|<math>\genexpintE{p}@{z} = \int_{1}^{\infty}\frac{e^{-zt}}{t^{p}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genexpintE{p}@{z} = \int_{1}^{\infty}\frac{e^{-zt}}{t^{p}}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>Ei(p, z) = int((exp(- z*t))/((t)^(p)), t = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ExpIntegralE[p, z] == Integrate[Divide[Exp[- z*t],(t)^(p)], {t, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 50]
|-
| [https://dlmf.nist.gov/8.19.E4 8.19.E4] || [[Item:Q2690|<math>\genexpintE{p}@{z} = \frac{z^{p-1}e^{-z}}{\EulerGamma@{p}}\int_{0}^{\infty}\frac{t^{p-1}e^{-zt}}{1+t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genexpintE{p}@{z} = \frac{z^{p-1}e^{-z}}{\EulerGamma@{p}}\int_{0}^{\infty}\frac{t^{p-1}e^{-zt}}{1+t}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{2}\pi, \realpart@@{p} > 0</math> || <syntaxhighlight lang=mathematica>Ei(p, z) = ((z)^(p - 1)* exp(- z))/(GAMMA(p))*int(((t)^(p - 1)* exp(- z*t))/(1 + t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ExpIntegralE[p, z] == Divide[(z)^(p - 1)* Exp[- z],Gamma[p]]*Integrate[Divide[(t)^(p - 1)* Exp[- z*t],1 + t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 25]
|-
| [https://dlmf.nist.gov/8.19.E5 8.19.E5] || [[Item:Q2691|<math>\genexpintE{0}@{z} = z^{-1}e^{-z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genexpintE{0}@{z} = z^{-1}e^{-z}</syntaxhighlight> || <math>z \neq 0</math> || <syntaxhighlight lang=mathematica>Ei(0, z) = (z)^(- 1)* exp(- z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ExpIntegralE[0, z] == (z)^(- 1)* Exp[- z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/8.19.E6 8.19.E6] || [[Item:Q2692|<math>\genexpintE{p}@{0} = \frac{1}{p-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genexpintE{p}@{0} = \frac{1}{p-1}</syntaxhighlight> || <math>\realpart@@{p} > 1</math> || <syntaxhighlight lang=mathematica>Ei(p, 0) = (1)/(p - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ExpIntegralE[p, 0] == Divide[1,p - 1]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 2]
|-
| [https://dlmf.nist.gov/8.19.E7 8.19.E7] || [[Item:Q2693|<math>\genexpintE{n}@{z} = \frac{(-z)^{n-1}}{(n-1)!}\expintE@{z}+\frac{e^{-z}}{(n-1)!}\sum_{k=0}^{n-2}(n-k-2)!(-z)^{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genexpintE{n}@{z} = \frac{(-z)^{n-1}}{(n-1)!}\expintE@{z}+\frac{e^{-z}}{(n-1)!}\sum_{k=0}^{n-2}(n-k-2)!(-z)^{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Ei(n, z) = ((- z)^(n - 1))/(factorial(n - 1))*Ei(z)+(exp(- z))/(factorial(n - 1))*sum(factorial(n - k - 2)*(- z)^(k), k = 0..n - 2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ExpIntegralE[n, z] == Divide[(- z)^(n - 1),(n - 1)!]*ExpIntegralE[1, z]+Divide[Exp[- z],(n - 1)!]*Sum[(n - k - 2)!*(- z)^(k), {k, 0, n - 2}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.393548628-1.498247032*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 1, n = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .4577249979+1.994294304*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 2, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/8.19.E9 8.19.E9] || [[Item:Q2695|<math>\genexpintE{n}@{z} = \frac{(-1)^{n}z^{n-1}}{(n-1)!}\ln@@{z}+\frac{e^{-z}}{(n-1)!}\sum_{k=1}^{n-1}(-z)^{k-1}\EulerGamma@{n-k}+\frac{e^{-z}(-z)^{n-1}}{(n-1)!}\sum_{k=0}^{\infty}\frac{z^{k}}{k!}\digamma@{k+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genexpintE{n}@{z} = \frac{(-1)^{n}z^{n-1}}{(n-1)!}\ln@@{z}+\frac{e^{-z}}{(n-1)!}\sum_{k=1}^{n-1}(-z)^{k-1}\EulerGamma@{n-k}+\frac{e^{-z}(-z)^{n-1}}{(n-1)!}\sum_{k=0}^{\infty}\frac{z^{k}}{k!}\digamma@{k+1}</syntaxhighlight> || <math>\realpart@@{n-k} > 0</math> || <syntaxhighlight lang=mathematica>Ei(n, z) = ((- 1)^(n)* (z)^(n - 1))/(factorial(n - 1))*ln(z)+(exp(- z))/(factorial(n - 1))*sum((- z)^(k - 1)* GAMMA(n - k), k = 1..n - 1)+(exp(- z)*(- z)^(n - 1))/(factorial(n - 1))*sum(((z)^(k))/(factorial(k))*Psi(k + 1), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ExpIntegralE[n, z] == Divide[(- 1)^(n)* (z)^(n - 1),(n - 1)!]*Log[z]+Divide[Exp[- z],(n - 1)!]*Sum[(- z)^(k - 1)* Gamma[n - k], {k, 1, n - 1}, GenerateConditions->None]+Divide[Exp[- z]*(- z)^(n - 1),(n - 1)!]*Sum[Divide[(z)^(k),(k)!]*PolyGamma[k + 1], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || <div class="toccolours mw-collapsible mw-collapsed">Failed [16 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.3691288000469654, -0.2016559825387078], Times[Complex[0.2188469268397846, -0.35920360372711485], Plus[-1.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Power[-1, Rational[1, 6]], []], Times[Plus[1, Times[-1, Power[-1, Rational[1, 6]]], Times[-1, ]], [Plus[1, ]]], Times[Plus[-1, ], [Plus[2, ]]]], 0], Equal[[0], 0], Equal[[1], 1]}]][2.0]]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.0256870546657635, -0.10579058942927923], Times[Complex[0.1094234634198923, -0.17960180186355743], Plus[-2.0, DifferenceRoot[Function[{, }
Test Values: {Equal[Plus[Times[Power[-1, Rational[1, 6]], []], Times[Plus[2, Times[-1, Power[-1, Rational[1, 6]]], Times[-1, ]], [Plus[1, ]]], Times[Plus[-2, ], [Plus[2, ]]]], 0], Equal[[0], 0], Equal[[1], 2]}]][3.0]]]], {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Ratio</div></div>
|-
| [https://dlmf.nist.gov/8.19.E10 8.19.E10] || [[Item:Q2696|<math>\genexpintE{p}@{z} = z^{p-1}\EulerGamma@{1-p}-\sum_{k=0}^{\infty}\frac{(-z)^{k}}{k!(1-p+k)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genexpintE{p}@{z} = z^{p-1}\EulerGamma@{1-p}-\sum_{k=0}^{\infty}\frac{(-z)^{k}}{k!(1-p+k)}</syntaxhighlight> || <math>\realpart@@{1-p} > 0</math> || <syntaxhighlight lang=mathematica>Ei(p, z) = (z)^(p - 1)* GAMMA(1 - p)- sum(((- z)^(k))/(factorial(k)*(1 - p + k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ExpIntegralE[p, z] == (z)^(p - 1)* Gamma[1 - p]- Sum[Divide[(- z)^(k),(k)!*(1 - p + k)], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 56]
|-
| [https://dlmf.nist.gov/8.19.E11 8.19.E11] || [[Item:Q2697|<math>\genexpintE{p}@{z} = \EulerGamma@{1-p}\left(z^{p-1}-e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{2-p+k}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genexpintE{p}@{z} = \EulerGamma@{1-p}\left(z^{p-1}-e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\EulerGamma@{2-p+k}}\right)</syntaxhighlight> || <math>\realpart@@{1-p} > 0, \realpart@@{2-p+k} > 0</math> || <syntaxhighlight lang=mathematica>Ei(p, z) = GAMMA(1 - p)*((z)^(p - 1)- exp(- z)*sum(((z)^(k))/(GAMMA(2 - p + k)), k = 0..infinity))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ExpIntegralE[p, z] == Gamma[1 - p]*((z)^(p - 1)- Exp[- z]*Sum[Divide[(z)^(k),Gamma[2 - p + k]], {k, 0, Infinity}, GenerateConditions->None])</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 56]
|-
| [https://dlmf.nist.gov/8.19.E12 8.19.E12] || [[Item:Q2698|<math>p\genexpintE{p+1}@{z}+z\genexpintE{p}@{z} = e^{-z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>p\genexpintE{p+1}@{z}+z\genexpintE{p}@{z} = e^{-z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>p*Ei(p + 1, z)+ z*Ei(p, z) = exp(- z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>p*ExpIntegralE[p + 1, z]+ z*ExpIntegralE[p, z] == Exp[- z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 70]
|-
| [https://dlmf.nist.gov/8.19.E13 8.19.E13] || [[Item:Q2699|<math>\deriv{}{z}\genexpintE{p}@{z} = -\genexpintE{p-1}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{z}\genexpintE{p}@{z} = -\genexpintE{p-1}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(Ei(p, z), z) = - Ei(p - 1, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[ExpIntegralE[p, z], z] == - ExpIntegralE[p - 1, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 70]
|-
| [https://dlmf.nist.gov/8.19.E14 8.19.E14] || [[Item:Q2700|<math>\deriv{}{z}(e^{z}\genexpintE{p}@{z}) = e^{z}\genexpintE{p}@{z}\left(1+\frac{p-1}{z}\right)-\frac{1}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{z}(e^{z}\genexpintE{p}@{z}) = e^{z}\genexpintE{p}@{z}\left(1+\frac{p-1}{z}\right)-\frac{1}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(exp(z)*Ei(p, z), z) = exp(z)*Ei(p, z)*(1 +(p - 1)/(z))-(1)/(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Exp[z]*ExpIntegralE[p, z], z] == Exp[z]*ExpIntegralE[p, z]*(1 +Divide[p - 1,z])-Divide[1,z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 70]
|-
| [https://dlmf.nist.gov/8.19.E15 8.19.E15] || [[Item:Q2701|<math>\pderiv[j]{\genexpintE{p}@{z}}{p} = (-1)^{j}\int_{1}^{\infty}(\ln@@{t})^{j}t^{-p}e^{-zt}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\pderiv[j]{\genexpintE{p}@{z}}{p} = (-1)^{j}\int_{1}^{\infty}(\ln@@{t})^{j}t^{-p}e^{-zt}\diff{t}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>diff(Ei(p, z), [p$(j)]) = (- 1)^(j)* int((ln(t))^(j)* (t)^(- p)* exp(- z*t), t = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[ExpIntegralE[p, z], {p, j}] == (- 1)^(j)* Integrate[(Log[t])^(j)* (t)^(- p)* Exp[- z*t], {t, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/8.19.E16 8.19.E16] || [[Item:Q2702|<math>\genexpintE{p}@{z} = z^{p-1}e^{-z}\KummerconfhyperU@{p}{p}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genexpintE{p}@{z} = z^{p-1}e^{-z}\KummerconfhyperU@{p}{p}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Ei(p, z) = (z)^(p - 1)* exp(- z)*KummerU(p, p, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ExpIntegralE[p, z] == (z)^(p - 1)* Exp[- z]*HypergeometricU[p, p, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 70]
|-
| [https://dlmf.nist.gov/8.19.E19 8.19.E19] || [[Item:Q2705|<math>\frac{n-1}{n}\genexpintE{n}@{x} < \genexpintE{n+1}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{n-1}{n}\genexpintE{n}@{x} < \genexpintE{n+1}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(n - 1)/(n)*Ei(n, x) < Ei(n + 1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[n - 1,n]*ExpIntegralE[n, x] < ExpIntegralE[n + 1, x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
|-
| [https://dlmf.nist.gov/8.19.E19 8.19.E19] || [[Item:Q2705|<math>\genexpintE{n+1}@{x} < \genexpintE{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genexpintE{n+1}@{x} < \genexpintE{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Ei(n + 1, x) < Ei(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ExpIntegralE[n + 1, x] < ExpIntegralE[n, x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
|-
| [https://dlmf.nist.gov/8.19.E20 8.19.E20] || [[Item:Q2706|<math>\left(\genexpintE{n}@{x}\right)^{2} < \genexpintE{n-1}@{x}\genexpintE{n+1}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\genexpintE{n}@{x}\right)^{2} < \genexpintE{n-1}@{x}\genexpintE{n+1}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(Ei(n, x))^(2) < Ei(n - 1, x)*Ei(n + 1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(ExpIntegralE[n, x])^(2) < ExpIntegralE[n - 1, x]*ExpIntegralE[n + 1, x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
|-
| [https://dlmf.nist.gov/8.19.E21 8.19.E21] || [[Item:Q2707|<math>\frac{1}{x+n} < e^{x}\genexpintE{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{x+n} < e^{x}\genexpintE{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(x + n) < exp(x)*Ei(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,x + n] < Exp[x]*ExpIntegralE[n, x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
|-
| [https://dlmf.nist.gov/8.19.E21 8.19.E21] || [[Item:Q2707|<math>e^{x}\genexpintE{n}@{x} \leq \frac{1}{x+n-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{x}\genexpintE{n}@{x} \leq \frac{1}{x+n-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>exp(x)*Ei(n, x) <= (1)/(x + n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[x]*ExpIntegralE[n, x] <= Divide[1,x + n - 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
|-
| [https://dlmf.nist.gov/8.19.E22 8.19.E22] || [[Item:Q2708|<math>\deriv{}{x}\frac{\genexpintE{n}@{x}}{\genexpintE{n-1}@{x}} > 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{x}\frac{\genexpintE{n}@{x}}{\genexpintE{n-1}@{x}} > 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff((Ei(n, x))/(Ei(n - 1, x)), x) > 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Divide[ExpIntegralE[n, x],ExpIntegralE[n - 1, x]], x] > 0</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
|-
| [https://dlmf.nist.gov/8.19.E23 8.19.E23] || [[Item:Q2709|<math>\int_{z}^{\infty}\genexpintE{p-1}@{t}\diff{t} = \genexpintE{p}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{z}^{\infty}\genexpintE{p-1}@{t}\diff{t} = \genexpintE{p}@{z}</syntaxhighlight> || <math>|\phase@@{z}| < \pi</math> || <syntaxhighlight lang=mathematica>int(Ei(p - 1, t), t = z..infinity) = Ei(p, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[ExpIntegralE[p - 1, t], {t, z, Infinity}, GenerateConditions->None] == ExpIntegralE[p, z]</syntaxhighlight> || Failure || Successful || Skipped - Because timed out || Successful [Tested: 70]
|-
| [https://dlmf.nist.gov/8.19.E24 8.19.E24] || [[Item:Q2710|<math>\int_{0}^{\infty}e^{-at}\genexpintE{n}@{t}\diff{t} = \frac{(-1)^{n-1}}{a^{n}}\left(\ln@{1+a}+\sum_{k=1}^{n-1}\frac{(-1)^{k}a^{k}}{k}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-at}\genexpintE{n}@{t}\diff{t} = \frac{(-1)^{n-1}}{a^{n}}\left(\ln@{1+a}+\sum_{k=1}^{n-1}\frac{(-1)^{k}a^{k}}{k}\right)</syntaxhighlight> || <math>\realpart@@{a} > -1</math> || <syntaxhighlight lang=mathematica>int(exp(- a*t)*Ei(n, t), t = 0..infinity) = ((- 1)^(n - 1))/((a)^(n))*(ln(1 + a)+ sum(((- 1)^(k)* (a)^(k))/(k), k = 1..n - 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- a*t]*ExpIntegralE[n, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[(- 1)^(n - 1),(a)^(n)]*(Log[1 + a]+ Sum[Divide[(- 1)^(k)* (a)^(k),k], {k, 1, n - 1}, GenerateConditions->None])</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 18]
|-
| [https://dlmf.nist.gov/8.19.E25 8.19.E25] || [[Item:Q2711|<math>\int_{0}^{\infty}e^{-at}t^{b-1}\genexpintE{p}@{t}\diff{t} = \frac{\EulerGamma@{b}(1+a)^{-b}}{p+b-1}\*\hyperF@{1}{b}{p+b}{a/(1+a)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-at}t^{b-1}\genexpintE{p}@{t}\diff{t} = \frac{\EulerGamma@{b}(1+a)^{-b}}{p+b-1}\*\hyperF@{1}{b}{p+b}{a/(1+a)}</syntaxhighlight> || <math>\realpart@@{a} > -1, \realpart@{p+b} > 1, \realpart@@{b} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- a*t)*(t)^(b - 1)* Ei(p, t), t = 0..infinity) = (GAMMA(b)*(1 + a)^(- b))/(p + b - 1)* hypergeom([1, b], [p + b], a/(1 + a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- a*t]*(t)^(b - 1)* ExpIntegralE[p, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[b]*(1 + a)^(- b),p + b - 1]* Hypergeometric2F1[1, b, p + b, a/(1 + a)]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Successful [Tested: 64]
|-
| [https://dlmf.nist.gov/8.19.E26 8.19.E26] || [[Item:Q2712|<math>\int_{0}^{\infty}\genexpintE{p}@{t}\genexpintE{q}@{t}\diff{t} = \frac{L(p)+L(q)}{p+q-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\genexpintE{p}@{t}\genexpintE{q}@{t}\diff{t} = \frac{L(p)+L(q)}{p+q-1}</syntaxhighlight> || <math>p > 0, q > 0, p+q > 1</math> || <syntaxhighlight lang=mathematica>int(Ei(p, t)*Ei(q, t), t = 0..infinity) = (L(p)+ L(q))/(p + q - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[ExpIntegralE[p, t]*ExpIntegralE[q, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[L[p]+ L[q],p + q - 1]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [80 / 80]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.8698344324715543, -0.7499999999999999]
Test Values: {Rule[L, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[p, 1.5], Rule[q, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.26794919243112303, -0.9999999999999999]
Test Values: {Rule[L, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[p, 1.5], Rule[q, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.19.E27 8.19.E27] || [[Item:Q2713|<math>L(p) = \int_{0}^{\infty}e^{-t}\genexpintE{p}@{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>L(p) = \int_{0}^{\infty}e^{-t}\genexpintE{p}@{t}\diff{t}</syntaxhighlight> || <math>p > 0</math> || <syntaxhighlight lang=mathematica>L(p) = int(exp(- t)*Ei(p, t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>L[p] == Integrate[Exp[- t]*ExpIntegralE[p, t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.8698344324715546, 0.7499999999999999]
Test Values: {Rule[L, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[p, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.1377836249026771, 0.24999999999999997]
Test Values: {Rule[L, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[p, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.19.E27 8.19.E27] || [[Item:Q2713|<math>\int_{0}^{\infty}e^{-t}\genexpintE{p}@{t}\diff{t} = \frac{1}{2p}\hyperF@{1}{1}{1+p}{\tfrac{1}{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-t}\genexpintE{p}@{t}\diff{t} = \frac{1}{2p}\hyperF@{1}{1}{1+p}{\tfrac{1}{2}}</syntaxhighlight> || <math>p > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- t)*Ei(p, t), t = 0..infinity) = (1)/(2*p)*hypergeom([1, 1], [1 + p], (1)/(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- t]*ExpIntegralE[p, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2*p]*Hypergeometric2F1[1, 1, 1 + p, Divide[1,2]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/8.20.E1 8.20.E1] || [[Item:Q2714|<math>\genexpintE{p}@{z} = \frac{e^{-z}}{z}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{\Pochhammersym{p}{k}}{z^{k}}+(-1)^{n}\frac{\Pochhammersym{p}{n}e^{z}}{z^{n-1}}\genexpintE{n+p}@{z}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\genexpintE{p}@{z} = \frac{e^{-z}}{z}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{\Pochhammersym{p}{k}}{z^{k}}+(-1)^{n}\frac{\Pochhammersym{p}{n}e^{z}}{z^{n-1}}\genexpintE{n+p}@{z}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Ei(p, z) = (exp(- z))/(z)*(sum((- 1)^(k)*(pochhammer(p, k))/((z)^(k)), k = 0..n - 1)+(- 1)^(n)*(pochhammer(p, n)*exp(z))/((z)^(n - 1))*Ei(n + p, z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ExpIntegralE[p, z] == Divide[Exp[- z],z]*(Sum[(- 1)^(k)*Divide[Pochhammer[p, k],(z)^(k)], {k, 0, n - 1}, GenerateConditions->None]+(- 1)^(n)*Divide[Pochhammer[p, n]*Exp[z],(z)^(n - 1)]*ExpIntegralE[n + p, z])</syntaxhighlight> || Failure || Successful || Manual Skip! || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[n, 3], Rule[p, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[n, 3], Rule[p, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.20.E4 8.20.E4] || [[Item:Q2717|<math>A_{k+1}(\lambda) = (1-2k\lambda)A_{k}(\lambda)+\lambda(\lambda+1)\deriv{A_{k}(\lambda)}{\lambda}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>A_{k+1}(\lambda) = (1-2k\lambda)A_{k}(\lambda)+\lambda(\lambda+1)\deriv{A_{k}(\lambda)}{\lambda}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>A[k + 1](lambda) = (1 - 2*k*lambda)*A[k](lambda)+ lambda*(lambda + 1)*diff(A[k](lambda), lambda)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[A, k + 1][\[Lambda]] == (1 - 2*k*\[Lambda])*Subscript[A, k][\[Lambda]]+ \[Lambda]*(\[Lambda]+ 1)*D[Subscript[A, k][\[Lambda]], \[Lambda]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.5000000000+.133974596*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, A[k] = 1/2*3^(1/2)+1/2*I, A[k+1] = 1/2*3^(1/2)+1/2*I, k = 1, k = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.4999999993+2.133974597*I
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, A[k] = 1/2*3^(1/2)+1/2*I, A[k+1] = 1/2*3^(1/2)+1/2*I, k = 2, k = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.49999999999999944, 4.133974596215561]
Test Values: {Rule[k, 3], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, Plus[1, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.8660254037844384, 3.7679491924311224]
Test Values: {Rule[k, 3], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[A, Plus[1, k]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/8.20#Ex1 8.20#Ex1] || [[Item:Q2718|<math>A_{1}(\lambda) = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>A_{1}(\lambda) = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>A[1](lambda) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[A, 1][\[Lambda]] == 1</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.20#Ex2 8.20#Ex2] || [[Item:Q2719|<math>A_{2}(\lambda) = 1-2\lambda</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>A_{2}(\lambda) = 1-2\lambda</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>A[2](lambda) = 1 - 2*lambda</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[A, 2][\[Lambda]] == 1 - 2*\[Lambda]</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.20#Ex3 8.20#Ex3] || [[Item:Q2720|<math>A_{3}(\lambda) = 1-8\lambda+6\lambda^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>A_{3}(\lambda) = 1-8\lambda+6\lambda^{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>A[3](lambda) = 1 - 8*lambda + 6*(lambda)^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[A, 3][\[Lambda]] == 1 - 8*\[Lambda]+ 6*\[Lambda]^(2)</syntaxhighlight> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/8.21.E3 8.21.E3] || [[Item:Q2724|<math>\int_{0}^{\infty}t^{a-1}e^{+\iunit t}\diff{t} = e^{+\frac{1}{2}\pi\iunit a}\EulerGamma@{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}t^{a-1}e^{+\iunit t}\diff{t} = e^{+\frac{1}{2}\pi\iunit a}\EulerGamma@{a}</syntaxhighlight> || <math>0 < \realpart@@{a}, \realpart@@{a} < 1, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^(a - 1)* exp(+ I*t), t = 0..infinity) = exp(+(1)/(2)*Pi*I*a)*GAMMA(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^(a - 1)* Exp[+ I*t], {t, 0, Infinity}, GenerateConditions->None] == Exp[+Divide[1,2]*Pi*I*a]*Gamma[a]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/8.21.E3 8.21.E3] || [[Item:Q2724|<math>\int_{0}^{\infty}t^{a-1}e^{-\iunit t}\diff{t} = e^{-\frac{1}{2}\pi\iunit a}\EulerGamma@{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}t^{a-1}e^{-\iunit t}\diff{t} = e^{-\frac{1}{2}\pi\iunit a}\EulerGamma@{a}</syntaxhighlight> || <math>0 < \realpart@@{a}, \realpart@@{a} < 1, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^(a - 1)* exp(- I*t), t = 0..infinity) = exp(-(1)/(2)*Pi*I*a)*GAMMA(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^(a - 1)* Exp[- I*t], {t, 0, Infinity}, GenerateConditions->None] == Exp[-Divide[1,2]*Pi*I*a]*Gamma[a]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/8.22.E1 8.22.E1] || [[Item:Q2750|<math>\frac{\EulerGamma@{p}}{2\pi}z^{1-p}\genexpintE{p}@{z} = \frac{\EulerGamma@{p}}{2\pi}\incGamma@{1-p}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\EulerGamma@{p}}{2\pi}z^{1-p}\genexpintE{p}@{z} = \frac{\EulerGamma@{p}}{2\pi}\incGamma@{1-p}{z}</syntaxhighlight> || <math>\realpart@@{p} > 0, \realpart@@{n+p} > 0</math> || <syntaxhighlight lang=mathematica>(GAMMA(p))/(2*Pi)*(z)^(1 - p)* Ei(p, z) = (GAMMA(p))/(2*Pi)*GAMMA(1 - p, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Gamma[p],2*Pi]*(z)^(1 - p)* ExpIntegralE[p, z] == Divide[Gamma[p],2*Pi]*Gamma[1 - p, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 35]
|-
| [https://dlmf.nist.gov/8.22.E3 8.22.E3] || [[Item:Q2752|<math>\zeta_{x}(s) = \sum_{k=1}^{\infty}k^{-s}\normincGammaP@{s}{kx}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\zeta_{x}(s) = \sum_{k=1}^{\infty}k^{-s}\normincGammaP@{s}{kx}</syntaxhighlight> || <math>\realpart@@{s} > 1</math> || <syntaxhighlight lang=mathematica>((1)/(GAMMA(s))*int(((t)^(s - 1))/(exp(t)- 1), t = 0..x)) = sum((k)^(- s)* (GAMMA(s)-GAMMA(s, k*x))/GAMMA(s), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Divide[1,Gamma[s]]*Integrate[Divide[(t)^(s - 1),Exp[t]- 1], {t, 0, x}, GenerateConditions->None]) == Sum[(k)^(- s)* GammaRegularized[s, 0, k*x], {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Manual Skip! || Skipped - Because timed out
|-
|}

Latest revision as of 16:47, 25 May 2021

Notation
8.1 Special Notation
Incomplete Gamma Functions
8.2 Definitions and Basic Properties
8.3 Graphics
8.4 Special Values
8.5 Confluent Hypergeometric Representations
8.6 Integral Representations
8.7 Series Expansions
8.8 Recurrence Relations and Derivatives
8.9 Continued Fractions
8.10 Inequalities
8.11 Asymptotic Approximations and Expansions
8.12 Uniform Asymptotic Expansions for Large Parameter
8.13 Zeros
8.14 Integrals
8.15 Sums
8.16 Generalizations
Related Functions
8.17 Incomplete Beta Functions
8.18 Asymptotic Expansions of Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \normincBetaI{x}@{a}{b}}
8.19 Generalized Exponential Integral
8.20 Asymptotic Expansions of Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \genexpintE{p}@{z}}
8.21 Generalized Sine and Cosine Integrals
Applications
8.22 Mathematical Applications
8.23 Statistical Applications
8.24 Physical Applications
Computation
8.25 Methods of Computation
8.26 Tables
8.27 Approximations
8.28 Software
Retrieved from "https://lct.wmflabs.org/w/index.php?title=Results_of_Incomplete_Gamma_and_Related_Functions&oldid=512"