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| | <div style="-moz-column-count:2; column-count:2;"> |
| |- | | ; Notation : [[8.1|8.1 Special Notation]]<br> |
| ! DLMF !! Formula !! 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>]] || <code>GAMMA(a)-GAMMA(a, z) = int((t)^(a - 1)* exp(- t), t = 0..z)</code> || <code>Gamma[a, 0, z] == Integrate[(t)^(a - 1)* Exp[- t], {t, 0, z}, GenerateConditions->None]</code> || 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>]] || <code>GAMMA(a, z) = int((t)^(a - 1)* exp(- t), t = z..infinity)</code> || <code>Gamma[a, z] == Integrate[(t)^(a - 1)* Exp[- t], {t, z, Infinity}, GenerateConditions->None]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [14 / 42]<div class="mw-collapsible-content"><code>14/42]: [[Float(infinity)+Float(infinity)*I <- {a = -2, z = 1/2*3^(1/2)+1/2*I}</code><br><code>Float(infinity)+Float(infinity)*I <- {a = -2, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || Successful [Tested: 42]
| | </div> |
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| | [https://dlmf.nist.gov/8.2.E3 8.2.E3] || [[Item:Q2482|<math>\incgamma@{a}{z}+\incGamma@{a}{z} = \EulerGamma@{a}</math>]] || <code>GAMMA(a)-GAMMA(a, z)+ GAMMA(a, z) = GAMMA(a)</code> || <code>Gamma[a, 0, z]+ Gamma[a, z] == Gamma[a]</code> || Successful || Successful || - || Successful [Tested: 7]
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| | [https://dlmf.nist.gov/8.2#Ex1 8.2#Ex1] || [[Item:Q2483|<math>\normincGammaP@{a}{z} = \frac{\incgamma@{a}{z}}{\EulerGamma@{a}}</math>]] || <code>(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = (GAMMA(a)-GAMMA(a, z))/(GAMMA(a))</code> || <code>GammaRegularized[a, 0, z] == Divide[Gamma[a, 0, z],Gamma[a]]</code> || Successful || Successful || - || Successful [Tested: 21]
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| |
| | [https://dlmf.nist.gov/8.2#Ex2 8.2#Ex2] || [[Item:Q2484|<math>\normincGammaQ@{a}{z} = \frac{\incGamma@{a}{z}}{\EulerGamma@{a}}</math>]] || <code>GAMMA(a, z)/GAMMA(a) = (GAMMA(a, z))/(GAMMA(a))</code> || <code>GammaRegularized[a, z] == Divide[Gamma[a, z],Gamma[a]]</code> || Successful || Successful || - || Successful [Tested: 21]
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| | [https://dlmf.nist.gov/8.2.E5 8.2.E5] || [[Item:Q2485|<math>\normincGammaP@{a}{z}+\normincGammaQ@{a}{z} = 1</math>]] || <code>(GAMMA(a)-GAMMA(a, z))/GAMMA(a)+ GAMMA(a, z)/GAMMA(a) = 1</code> || <code>GammaRegularized[a, 0, z]+ GammaRegularized[a, z] == 1</code> || Successful || Successful || - || Successful [Tested: 21]
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| | [https://dlmf.nist.gov/8.2.E6 8.2.E6] || [[Item:Q2486|<math>\scincgamma@{a}{z} = z^{-a}\normincGammaP@{a}{z}</math>]] || <code>(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = (z)^(- a)* (GAMMA(a)-GAMMA(a, z))/GAMMA(a)</code> || <code>Error</code> || Successful || Missing Macro Error || Skip - symbolical successful subtest || -
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| | [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>]] || <code>(z)^(- a)* (GAMMA(a)-GAMMA(a, z))/GAMMA(a) = ((z)^(- a))/(GAMMA(a))*GAMMA(a)-GAMMA(a, z)</code> || <code>(z)^(- a)* GammaRegularized[a, 0, z] == Divide[(z)^(- a),Gamma[a]]*Gamma[a, 0, z]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><code>21/21]: [[.2669922311+.3869142026*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I}</code><br><code>3.493684189-.9843316111*I <- {a = 1.5, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || Successful [Tested: 21]
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| | [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>]] || <code>(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = (1)/(GAMMA(a))*int((t)^(a - 1)* exp(- z*t), t = 0..1)</code> || <code>Error</code> || Failure || Missing Macro Error || Successful [Tested: 21] || -
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| | [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>]] || <code>GAMMA(a)-GAMMA(a, z*exp(2*Pi*m*I)) = exp(2*Pi*m*I*a)*GAMMA(a)-GAMMA(a, z)</code> || <code>Gamma[a, 0, z*Exp[2*Pi*m*I]] == Exp[2*Pi*m*I*a]*Gamma[a, 0, z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [28 / 63]<div class="mw-collapsible-content"><code>28/63]: [[1.772453851+.14e-8*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I, m = 1, a = 1}</code><br><code>1.772453851+.62e-8*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I, m = 3, a = 1}</code><br></div></div> || Successful [Tested: 21]
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| | [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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [28 / 63]<div class="mw-collapsible-content"><code>28/63]: [[-.701776495-.3980325655*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I, m = 1}</code><br><code>-.701776493-.3980325669*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I, m = 3}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [28 / 63]<div class="mw-collapsible-content"><code>{Complex[-0.7017764946044596, -0.3980325648566406] <- {Rule[a, 1.5], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-0.7017764946044598, -0.39803256485664035] <- {Rule[a, 1.5], Rule[m, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [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>]] || <code>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))</code> || <code>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]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [28 / 28]<div class="mw-collapsible-content"><code>28/28]: [[-2.710645106+2.690107924*I <- {a = -1.5, z = 1/2*3^(1/2)+1/2*I}</code><br><code>.1295806364+5.171352915*I <- {a = -1.5, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [28 / 28]<div class="mw-collapsible-content"><code>{Complex[-2.710645107283499, 2.690107923521634] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[0.12958063668571548, 5.171352913200156] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [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>]] || <code>GAMMA(a, z*exp(+ Pi*I)) = GAMMA(a)*(1 - (z)^(a)* exp(+ Pi*I*a)*(- z)^(-(a))*(GAMMA(a)-GAMMA(a, - z))/GAMMA(a))</code> || <code>Error</code> || Failure || Missing Macro Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 21]<div class="mw-collapsible-content"><code>4/21]: [[2.046852240-1.006153525*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I}</code><br><code>-.4563214597+.8560373719*I <- {a = 1.5, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || -
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| | [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>]] || <code>GAMMA(a, z*exp(- Pi*I)) = GAMMA(a)*(1 - (z)^(a)* exp(- Pi*I*a)*(- z)^(-(a))*(GAMMA(a)-GAMMA(a, - z))/GAMMA(a))</code> || <code>Error</code> || Failure || Missing Macro Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 21]<div class="mw-collapsible-content"><code>10/21]: [[1.510498964-.9249588863*I <- {a = 1.5, z = 1/2-1/2*I*3^(1/2)}</code><br><code>-.7017764949-.3980325647*I <- {a = 1.5, z = -1/2*3^(1/2)-1/2*I}</code><br></div></div> || -
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| | [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>]] || <code>diff(w, [z$(2)])+(1 +(1 - a)/(z))* diff(w, z) = 0</code> || <code>D[w, {z, 2}]+(1 +Divide[1 - a,z])* D[w, z] == 0</code> || Successful || Successful || - || Successful [Tested: 300]
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| | [https://dlmf.nist.gov/8.2.E13 8.2.E13] || [[Item:Q2493|<math>\deriv[2]{w}{z}-\left(1+\frac{1-a}{z}\right)\deriv{w}{z}+\frac{1-a}{z^{2}}w = 0</math>]] || <code>diff(w, [z$(2)])-(1 +(1 - a)/(z))* diff(w, z)+(1 - a)/((z)^(2))*w = 0</code> || <code>D[w, {z, 2}]-(1 +Divide[1 - a,z])* D[w, z]+Divide[1 - a,(z)^(2)]*w == 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[2.165063509-1.250000000*I <- {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</code><br><code>-2.165063509+1.250000000*I <- {a = -1.5, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[2.1650635094610964, -1.25] <- {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]]]}</code><br><code>Complex[-2.1650635094610964, 1.2500000000000004] <- {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]]]}</code><br></div></div>
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| | [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>]] || <code>GAMMA((1)/(2))-GAMMA((1)/(2), (z)^(2)) = 2*int(exp(- (t)^(2)), t = 0..z)</code> || <code>Gamma[Divide[1,2], 0, (z)^(2)] == 2*Integrate[Exp[- (t)^(2)], {t, 0, z}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><code>2/7]: [[3.465949776-3.038201708*I <- {z = -1/2+1/2*I*3^(1/2)}</code><br><code>3.197911286+.8974462698*I <- {z = -1/2*3^(1/2)-1/2*I}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><code>{Complex[3.4659497742269214, -3.038201707267986] <- {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br><code>Complex[3.197911285535813, 0.8974462701863266] <- {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</code><br></div></div>
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| | [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>]] || <code>2*int(exp(- (t)^(2)), t = 0..z) = sqrt(Pi)*erf(z)</code> || <code>2*Integrate[Exp[- (t)^(2)], {t, 0, z}, GenerateConditions->None] == Sqrt[Pi]*Erf[z]</code> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 7]
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| | [https://dlmf.nist.gov/8.4.E2 8.4.E2] || [[Item:Q2496|<math>\scincgamma@{a}{0} = \frac{1}{\EulerGamma@{a+1}}</math>]] || <code>(0)^(-(a))*(GAMMA(a)-GAMMA(a, 0))/GAMMA(a) = (1)/(GAMMA(a + 1))</code> || <code>Error</code> || Failure || Missing Macro Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>3/3]: [[-.7522527782 <- {a = 1.5}</code><br><code>-1.128379167 <- {a = .5}</code><br></div></div> || -
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| | [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>]] || <code>(- (z)^(2))^(-((1)/(2)))*(GAMMA((1)/(2))-GAMMA((1)/(2), - (z)^(2)))/GAMMA((1)/(2)) = (2*exp((z)^(2)))/(z*sqrt(Pi))*dawson(z)</code> || <code>Error</code> || Successful || Missing Macro Error || - || -
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| | [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>]] || <code>GAMMA(0, z) = int((t)^(- 1)* exp(- t), t = z..infinity)</code> || <code>Gamma[0, z] == Integrate[(t)^(- 1)* Exp[- t], {t, z, Infinity}, GenerateConditions->None]</code> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 7]
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| | [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>]] || <code>int((t)^(- 1)* exp(- t), t = z..infinity) = Ei(z)</code> || <code>Integrate[(t)^(- 1)* Exp[- t], {t, z, Infinity}, GenerateConditions->None] == ExpIntegralE[1, z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><code>7/7]: [[-1.393548628-1.498247032*I <- {z = 1/2*3^(1/2)+1/2*I}</code><br><code>-.8944744989-3.773814377*I <- {z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || Successful [Tested: 7]
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| | [https://dlmf.nist.gov/8.4.E5 8.4.E5] || [[Item:Q2499|<math>\incGamma@{1}{z} = e^{-z}</math>]] || <code>GAMMA(1, z) = exp(- z)</code> || <code>Gamma[1, z] == Exp[- z]</code> || Successful || Successful || - || Successful [Tested: 7]
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| | [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>]] || <code>GAMMA((1)/(2), (z)^(2)) = 2*int(exp(- (t)^(2)), t = z..infinity)</code> || <code>Gamma[Divide[1,2], (z)^(2)] == 2*Integrate[Exp[- (t)^(2)], {t, z, Infinity}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><code>2/7]: [[-3.465949776+3.038201708*I <- {z = -1/2+1/2*I*3^(1/2)}</code><br><code>-3.197911286-.8974462698*I <- {z = -1/2*3^(1/2)-1/2*I}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><code>{Complex[-3.4659497742269214, 3.038201707267986] <- {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br><code>Complex[-3.1979112855358127, -0.8974462701863266] <- {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</code><br></div></div>
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| | [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>]] || <code>2*int(exp(- (t)^(2)), t = z..infinity) = sqrt(Pi)*erfc(z)</code> || <code>2*Integrate[Exp[- (t)^(2)], {t, z, Infinity}, GenerateConditions->None] == Sqrt[Pi]*Erfc[z]</code> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 7]
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| | [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>]] || <code>GAMMA(n + 1)-GAMMA(n + 1, z) = factorial(n)*(1 - exp(- z)*exp(1)[n]*(z))</code> || <code>Gamma[n + 1, 0, z] == (n)!*(1 - Exp[- z]*Subscript[E, n]*(z))</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><code>{Plus[Complex[-0.7896317094254578, 0.19173078621885742], Times[Complex[0.42050290937849244, 0.009925196319850484], Subscript[2.718281828459045, 1]]] <- {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Complex[0.06153297742196945, 0.16461464559793018], Times[-2.0, Plus[1.0, Times[Complex[-0.42050290937849244, -0.009925196319850484], Subscript[2.718281828459045, 2]]]]] <- {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/8.4.E8 8.4.E8] || [[Item:Q2502|<math>\incGamma@{n+1}{z} = n!e^{-z}e_{n}(z)</math>]] || <code>GAMMA(n + 1, z) = factorial(n)*exp(- z)*exp(1)[n]*(z)</code> || <code>Gamma[n + 1, z] == (n)!*Exp[- z]*Subscript[E, n]*(z)</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><code>{Plus[Complex[0.7896317094254578, -0.19173078621885742], Times[Complex[-0.42050290937849244, -0.009925196319850484], Subscript[2.718281828459045, 1]]] <- {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Complex[1.9384670225780305, -0.16461464559793018], Times[Complex[-0.8410058187569849, -0.019850392639700967], Subscript[2.718281828459045, 2]]] <- {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></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>]] || <code>(GAMMA(n + 1)-GAMMA(n + 1, z))/GAMMA(n + 1) = 1 - exp(- z)*exp(1)[n]*(z)</code> || <code>GammaRegularized[n + 1, 0, z] == 1 - Exp[- z]*Subscript[E, n]*(z)</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><code>{Plus[Complex[-0.7896317094254579, 0.1917307862188573], Times[Complex[0.42050290937849244, 0.009925196319850484], Subscript[2.718281828459045, 1]]] <- {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Complex[-0.9692335112890152, 0.08230732279896512], Times[Complex[0.42050290937849244, 0.009925196319850484], Subscript[2.718281828459045, 2]]] <- {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></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>]] || <code>GAMMA(n + 1, z)/GAMMA(n + 1) = exp(- z)*exp(1)[n]*(z)</code> || <code>GammaRegularized[n + 1, z] == Exp[- z]*Subscript[E, n]*(z)</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><code>{Plus[Complex[0.7896317094254579, -0.1917307862188573], Times[Complex[-0.42050290937849244, -0.009925196319850484], Subscript[2.718281828459045, 1]]] <- {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Complex[0.9692335112890152, -0.08230732279896512], Times[Complex[-0.42050290937849244, -0.009925196319850484], Subscript[2.718281828459045, 2]]] <- {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/8.4.E12 8.4.E12] || [[Item:Q2506|<math>\scincgamma@{-n}{z} = z^{n}</math>]] || <code>(z)^(-(- n))*(GAMMA(- n)-GAMMA(- n, z))/GAMMA(- n) = (z)^(n)</code> || <code>Error</code> || 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>]] || <code>GAMMA(1 - n, z) = (z)^(1 - n)* Ei(n, z)</code> || <code>Gamma[1 - n, z] == (z)^(1 - n)* ExpIntegralE[n, z]</code> || 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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><code>6/21]: [[1.704415567+1.043704337*I <- {z = -1/2+1/2*I*3^(1/2), n = 1}</code><br><code>.97393781e-1-.8458491548*I <- {z = -1/2+1/2*I*3^(1/2), n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><code>{Complex[1.7044155650581054, 1.0437043365740406] <- {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br><code>Complex[0.09739377924871273, -0.8458491528064774] <- {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></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>]] || <code>GAMMA(- n, z) = ((- 1)^(n))/(factorial(n))*(Ei(z)- exp(- z)*sum(((- 1)^(k)* factorial(k))/((z)^(k + 1)), k = 0..n - 1))</code> || <code>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])</code> || 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>]] || <code>GAMMA(a)-GAMMA(a, z) = (a)^(- 1)* (z)^(a)* exp(- z)*KummerM(1, 1 + a, z)</code> || <code>Gamma[a, 0, z] == (a)^(- 1)* (z)^(a)* Exp[- z]*Hypergeometric1F1[1, 1 + a, z]</code> || 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>]] || <code>(a)^(- 1)* (z)^(a)* exp(- z)*KummerM(1, 1 + a, z) = (a)^(- 1)* (z)^(a)* KummerM(a, 1 + a, - z)</code> || <code>(a)^(- 1)* (z)^(a)* Exp[- z]*Hypergeometric1F1[1, 1 + a, z] == (a)^(- 1)* (z)^(a)* Hypergeometric1F1[a, 1 + a, - z]</code> || 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>]] || <code>(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = exp(- z)*KummerM(1, 1 + a, z)/GAMMA(1 + a)</code> || <code>Error</code> || 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>]] || <code>exp(- z)*KummerM(1, 1 + a, z)/GAMMA(1 + a) = KummerM(a, 1 + a, - z)/GAMMA(1 + a)</code> || <code>Exp[- z]*Hypergeometric1F1Regularized[1, 1 + a, z] == Hypergeometric1F1Regularized[a, 1 + a, - z]</code> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 42]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Indeterminate <- {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></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>]] || <code>GAMMA(a, z) = exp(- z)*KummerU(1 - a, 1 - a, z)</code> || <code>Gamma[a, z] == Exp[- z]*HypergeometricU[1 - a, 1 - a, z]</code> || 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>]] || <code>exp(- z)*KummerU(1 - a, 1 - a, z) = (z)^(a)* exp(- z)*KummerU(1, 1 + a, z)</code> || <code>Exp[- z]*HypergeometricU[1 - a, 1 - a, z] == (z)^(a)* Exp[- z]*HypergeometricU[1, 1 + a, z]</code> || 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>]] || <code>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)</code> || <code>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]</code> || 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>]] || <code>GAMMA(a, z) = exp(-(1)/(2)*z)*(z)^((1)/(2)*a -(1)/(2))* WhittakerW((1)/(2)*a -(1)/(2), (1)/(2)*a, z)</code> || <code>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]</code> || 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>]] || <code>GAMMA(a)-GAMMA(a, z) = ((z)^(a))/(sin(Pi*a))*int(exp(z*cos(t))*cos(a*t + z*sin(t)), t = 0..Pi)</code> || <code>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]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [14 / 21]<div class="mw-collapsible-content"><code>14/21]: [[1.922649672+.1964472815*I <- {a = .5, z = 1/2*3^(1/2)+1/2*I, a = 3/2}</code><br><code>2.511118576+1.941926371*I <- {a = .5, z = -1/2+1/2*I*3^(1/2), a = 3/2}</code><br></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>]] || <code>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)</code> || <code>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]</code> || 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>]] || <code>GAMMA(a)-GAMMA(a, z) = (z)^(a)* int(exp(- a*t - z*exp(- t)), t = 0..infinity)</code> || <code>Gamma[a, 0, z] == (z)^(a)* Integrate[Exp[- a*t - z*Exp[- t]], {t, 0, Infinity}, GenerateConditions->None]</code> || 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>]] || <code>GAMMA(a, z) = ((z)^(a)* exp(- z))/(GAMMA(1 - a))*int(((t)^(- a)* exp(- t))/(z + t), t = 0..infinity)</code> || <code>Gamma[a, z] == Divide[(z)^(a)* Exp[- z],Gamma[1 - a]]*Integrate[Divide[(t)^(- a)* Exp[- t],z + t], {t, 0, Infinity}, GenerateConditions->None]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 28]<div class="mw-collapsible-content"><code>12/28]: [[Float(infinity)+Float(infinity)*I <- {a = -1.5, z = 1/2*3^(1/2)+1/2*I}</code><br><code>Float(infinity)+Float(infinity)*I <- {a = -1.5, z = -1/2+1/2*I*3^(1/2)}</code><br></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>]] || <code>GAMMA(a, z) = (z)^(a)* exp(- z)*int((exp(- z*t))/((1 + t)^(1 - a)), t = 0..infinity)</code> || <code>Gamma[a, z] == (z)^(a)* Exp[- z]*Integrate[Divide[Exp[- z*t],(1 + t)^(1 - a)], {t, 0, Infinity}, GenerateConditions->None]</code> || 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>]] || <code>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)</code> || <code>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]</code> || 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>]] || <code>GAMMA(a, z) = (z)^(a)* int(exp(a*t - z*exp(t)), t = 0..infinity)</code> || <code>Gamma[a, z] == (z)^(a)* Integrate[Exp[a*t - z*Exp[t]], {t, 0, Infinity}, GenerateConditions->None]</code> || 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>]] || <code>GAMMA(a)-GAMMA(a, z) = (- I*(z)^(a))/(2*sin(Pi*a))*int((t)^(a - 1)* exp(z*t), t = - 1..(0 +))</code> || <code>Gamma[a, 0, z] == Divide[- I*(z)^(a),2*Sin[Pi*a]]*Integrate[(t)^(a - 1)* Exp[z*t], {t, - 1, (0 +)}, GenerateConditions->None]</code> || 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>]] || <code>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)</code> || <code>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]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 20]<div class="mw-collapsible-content"><code>20/20]: [[Float(infinity)+Float(infinity)*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I}</code><br><code>Float(infinity)+Float(infinity)*I <- {a = 1.5, z = 1/2-1/2*I*3^(1/2)}</code><br></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>]] || <code>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)</code> || <code>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]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 20]<div class="mw-collapsible-content"><code>20/20]: [[Float(infinity)+Float(infinity)*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I}</code><br><code>Float(infinity)+Float(infinity)*I <- {a = 1.5, z = 1/2-1/2*I*3^(1/2)}</code><br></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>]] || <code>GAMMA(a)-GAMMA(a, z) = (1)/(2*Pi*I)*int((GAMMA(s))/(a - s)*(z)^(a - s), s = c - I*infinity..c + I*infinity)</code> || <code>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]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [90 / 90]<div class="mw-collapsible-content"><code>90/90]: [[.3508882474+.1990162824*I <- {a = 1.5, c = -1.5, z = 1/2*3^(1/2)+1/2*I, a = 1}</code><br><code>.2281607298-.4280186861*I <- {a = 1.5, c = -1.5, z = 1/2-1/2*I*3^(1/2), a = 1}</code><br></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>]] || <code>GAMMA(a, z) = (1)/(2*Pi*I)*int(GAMMA(s + a)*((z)^(- s))/(s), s = c - I*infinity..c + I*infinity)</code> || <code>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]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><code>180/180]: [[.1072320848e-1-.1480251451*I <- {a = -1.5, c = -1.5, z = 1/2*3^(1/2)+1/2*I}</code><br><code>-.2224046553+.6479031822e-1*I <- {a = -1.5, c = -1.5, z = 1/2-1/2*I*3^(1/2)}</code><br></div></div> || Skipped - Because timed out
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| |-
| |
| | [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>]] || <code>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)</code> || <code>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]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [168 / 168]<div class="mw-collapsible-content"><code>168/168]: [[.1072320848e-1-.1480251451*I <- {a = -1.5, c = -1.5, z = 1/2*3^(1/2)+1/2*I, a = -1}</code><br><code>.7867555591e-1+.8824866094*I <- {a = -1.5, c = -1.5, z = -1/2+1/2*I*3^(1/2), a = -1}</code><br></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>]] || <code>(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = exp(- z)*sum(((z)^(k))/(GAMMA(a + k + 1)), k = 0..infinity)</code> || <code>Error</code> || 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>]] || <code>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)</code> || <code>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]</code> || 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>]] || <code>GAMMA(a)-GAMMA(a, x + y)- GAMMA(a)-GAMMA(a, x) = GAMMA(a, x)- GAMMA(a, x + y)</code> || <code>Gamma[a, 0, x + y]- Gamma[a, 0, x] == Gamma[a, x]- Gamma[a, x + y]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><code>18/18]: [[-.6941375518 <- {a = 1.5, x = 1.5, y = -.5}</code><br><code>-.6941375518 <- {a = 1.5, x = 1.5, y = .5}</code><br></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>]] || <code>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)</code> || <code>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]</code> || 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>]] || <code>GAMMA(a, z) = GAMMA(a)- sum(((- 1)^(k)* (z)^(a + k))/(factorial(k)*(a + k)), k = 0..infinity)</code> || <code>Gamma[a, z] == Gamma[a]- Sum[Divide[(- 1)^(k)* (z)^(a + k),(k)!*(a + k)], {k, 0, Infinity}, GenerateConditions->None]</code> || 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>]] || <code>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))</code> || <code>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])</code> || Successful || Successful || - || Successful [Tested: 7]
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| |-
| |
| | [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>]] || <code>Error</code> || <code>Gamma[a, x] == (x)^(a)* Exp[- x]*Sum[Divide[LaguerreL[n, a, x],n + 1], {n, 0, Infinity}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><code>{Plus[0.03483445061608508, Times[-0.1214566752420326, NSum[Times[Power[Plus[1, n], -1], LaguerreL[n, -1.5, 1.5]] <- {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[a, -1.5], Rule[x, 1.5]}</code><br><code>Plus[0.7498909754592095, Times[-1.7155277699214138, NSum[Times[Power[Plus[1, n], -1], LaguerreL[n, -1.5, 0.5]] <- {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[a, -1.5], Rule[x, 0.5]}</code><br></div></div>
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| |-
| |
| | [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>]] || <code>GAMMA(a + 1)-GAMMA(a + 1, z) = a*GAMMA(a)-GAMMA(a, z)- (z)^(a)* exp(- z)</code> || <code>Gamma[a + 1, 0, z] == a*Gamma[a, 0, z]- (z)^(a)* Exp[- z]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><code>21/21]: [[-.2676693395+.995081412e-1*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I}</code><br><code>-.820738205+.231239721*I <- {a = 1.5, z = -1/2+1/2*I*3^(1/2)}</code><br></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>]] || <code>GAMMA(a + 1, z) = a*GAMMA(a, z)+ (z)^(a)* exp(- z)</code> || <code>Gamma[a + 1, z] == a*Gamma[a, z]+ (z)^(a)* Exp[- z]</code> || 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>]] || <code>w*(a + 2 , z)-(a + 1 + z)* w*(a + 1 , z)+ a*z*w*(a , z) = 0</code> || <code>w*(a + 2 , z)-(a + 1 + z)* w*(a + 1 , z)+ a*z*w*(a , z) == 0</code> || 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>]] || <code>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))</code> || <code>Error</code> || Failure || Missing Macro Error || Successful [Tested: 21] || -
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| |-
| |
| | [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>]] || <code>(GAMMA(a + 1)-GAMMA(a + 1, z))/GAMMA(a + 1) = (GAMMA(a)-GAMMA(a, z))/GAMMA(a)-((z)^(a)* exp(- z))/(GAMMA(a + 1))</code> || <code>GammaRegularized[a + 1, 0, z] == GammaRegularized[a, 0, z]-Divide[(z)^(a)* Exp[- z],Gamma[a + 1]]</code> || 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>]] || <code>GAMMA(a + 1, z)/GAMMA(a + 1) = GAMMA(a, z)/GAMMA(a)+((z)^(a)* exp(- z))/(GAMMA(a + 1))</code> || <code>GammaRegularized[a + 1, z] == GammaRegularized[a, z]+Divide[(z)^(a)* Exp[- z],Gamma[a + 1]]</code> || Failure || Successful || Successful [Tested: 28] || Successful [Tested: 21]
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| |-
| |
| | [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>]] || <code>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)</code> || <code>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]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [63 / 63]<div class="mw-collapsible-content"><code>63/63]: [[-.2676693391+.995081412e-1*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 1}</code><br><code>-1.472181365+.5472947763*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 2}</code><br></div></div> || Successful [Tested: 63]
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| |-
| |
| | [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>]] || <code>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)</code> || <code>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]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 14]<div class="mw-collapsible-content"><code>7/14]: [[.1265952281-.9976912441e-1*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 1}</code><br><code>.19739482e-1-.7595504274*I <- {a = 1.5, z = -1/2+1/2*I*3^(1/2), n = 1}</code><br></div></div> || Successful [Tested: 14]
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| |-
| |
| | [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>]] || <code>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)</code> || <code>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]</code> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 105]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Indeterminate <- {Rule[a, -2], Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [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>]] || <code>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)</code> || <code>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]</code> || Failure || Successful || Successful [Tested: 14] || Successful [Tested: 14]
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| |-
| |
| | [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>]] || <code>(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)</code> || <code>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]</code> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 126]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[a, -2], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Indeterminate <- {Rule[a, -2], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| |-
| |
| | [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>]] || <code>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)</code> || <code>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]</code> || Successful || Successful || - || Successful [Tested: 63]
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| |-
| |
| | [https://dlmf.nist.gov/8.8.E13 8.8.E13] || [[Item:Q2545|<math>\deriv{}{z}\incgamma@{a}{z} = -\deriv{}{z}\incGamma@{a}{z}</math>]] || <code>diff(GAMMA(a)-GAMMA(a, z), z) = - diff(GAMMA(a, z), z)</code> || <code>D[Gamma[a, 0, z], z] == - D[Gamma[a, z], z]</code> || Successful || Successful || - || Successful [Tested: 21]
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| |-
| |
| | [https://dlmf.nist.gov/8.8.E13 8.8.E13] || [[Item:Q2545|<math>-\deriv{}{z}\incGamma@{a}{z} = z^{a-1}e^{-z}</math>]] || <code>- diff(GAMMA(a, z), z) = (z)^(a - 1)* exp(- z)</code> || <code>- D[Gamma[a, z], z] == (z)^(a - 1)* Exp[- z]</code> || Successful || Successful || - || Successful [Tested: 21]
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| |-
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| | [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>]] || <code>diff((z)^(- a)* GAMMA(a)-GAMMA(a, z), [z$(n)]) = (- 1)^(n)* (z)^(- a - n)* GAMMA(a + n)-GAMMA(a + n, z)</code> || <code>D[(z)^(- a)* Gamma[a, 0, z], {z, n}] == (- 1)^(n)* (z)^(- a - n)* Gamma[a + n, 0, z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [63 / 63]<div class="mw-collapsible-content"><code>63/63]: [[1.615357258-.2793504168*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 1}</code><br><code>3.050292670-.1918135000*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [63 / 63]<div class="mw-collapsible-content"><code>{Plus[Complex[0.20573036539123668, -0.07193062032175179], Inactive[Sum][Times[Power[Complex[0.8660254037844387, 0.49999999999999994], Plus[-1.5, Times[-1.0, K[1.0]]]], Binomial[1.0, K[1.0]], D[Complex[0.3508882473022298, 0.19901628242832037] <- {Complex[0.8660254037844387, 0.49999999999999994], Plus[1.0, Times[-1.0, K[1.0]]]}], FactorialPower[-1.5, K[1.0]]], {K[1.0], 0.0, 1.0}]], {Rule[a, 1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Complex[-0.13665910465469025, 0.05369371428345661], Inactive[Sum][Times[Power[Complex[0.8660254037844387, 0.49999999999999994], Plus[-1.5, Times[-1.0, K[1.0]]]], Binomial[2.0, K[1.0]], D[Complex[0.3508882473022298, 0.19901628242832037] <- {Complex[0.8660254037844387, 0.49999999999999994], Plus[2.0, Times[-1.0, K[1.0]]]}], FactorialPower[-1.5, K[1.0]]], {K[1.0], 0.0, 2.0}]], {Rule[a, 1.5], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/8.8.E16 8.8.E16] || [[Item:Q2548|<math>\deriv[n]{}{z}(z^{-a}\incGamma@{a}{z}) = (-1)^{n}z^{-a-n}\incGamma@{a+n}{z}</math>]] || <code>diff((z)^(- a)* GAMMA(a, z), [z$(n)]) = (- 1)^(n)* (z)^(- a - n)* GAMMA(a + n, z)</code> || <code>D[(z)^(- a)* Gamma[a, z], {z, n}] == (- 1)^(n)* (z)^(- a - n)* Gamma[a + n, z]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [111 / 126]<div class="mw-collapsible-content"><code>{Plus[Complex[0.14584260074790834, -0.14889469354125948], Times[Complex[0.9659258262890683, 0.25881904510252074], DifferenceRoot[Function[{, } <- {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]], Pi]]], Times[-2, , 1, Power[E, Times[Complex</div></div>
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| |-
| |
| | [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>]] || <code>diff(exp(z)*GAMMA(a)-GAMMA(a, z), [z$(n)]) = (- 1)^(n)* pochhammer(1 - a, n)*exp(z)*GAMMA(a - n)-GAMMA(a - n, z)</code> || <code>D[Exp[z]*Gamma[a, 0, z], {z, n}] == (- 1)^(n)* Pochhammer[1 - a, n]*Exp[z]*Gamma[a - n, 0, z]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [14 / 14]<div class="mw-collapsible-content"><code>14/14]: [[.6619339064-.2987854069*I <- {a = 1.5, z = 1/2*3^(1/2)+1/2*I, n = 1}</code><br><code>1.661215891-1.222029870*I <- {a = 1.5, z = -1/2+1/2*I*3^(1/2), n = 1}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [14 / 14]<div class="mw-collapsible-content"><code>{Plus[Complex[-1.471179750131411, -1.0739918488339026], Inactive[Sum][Times[Complex[2.0864022336812553, 1.1398067350757155], Binomial[1.0, K[1.0]], D[Complex[0.3508882473022298, 0.19901628242832037] <- {Complex[0.8660254037844387, 0.49999999999999994], Plus[1.0, Times[-1.0, K[1.0]]]}]], {K[1.0], 0.0, 1.0}]], {Rule[a, 1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Complex[0.01045242262446705, -0.698806597134537], Inactive[Sum][Times[Complex[0.3929465558343552, 0.4620307840711054], Binomial[1.0, K[1.0]], D[Complex[-0.7552494829576352, 0.46247944264186114] <- {Complex[-0.4999999999999998, 0.8660254037844387], Plus[1.0, Times[-1.0, K[1.0]]]}]], {K[1.0], 0.0, 1.0}]], {Rule[a, 1.5], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/8.8.E18 8.8.E18] || [[Item:Q2550|<math>\deriv[n]{}{z}(z^{a}e^{z}\scincgamma@{a}{z}) = z^{a-n}e^{z}\scincgamma@{a-n}{z}</math>]] || <code>diff((z)^(a)* exp(z)*(z)^(-(a))*(GAMMA(a)-GAMMA(a, z))/GAMMA(a), [z$(n)]) = (z)^(a - n)* exp(z)*(z)^(-(a - n))*(GAMMA(a - n)-GAMMA(a - n, z))/GAMMA(a - n)</code> || <code>Error</code> || 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>]] || <code>diff(exp(z)*GAMMA(a, z), [z$(n)]) = (- 1)^(n)* pochhammer(1 - a, n)*exp(z)*GAMMA(a - n, z)</code> || <code>D[Exp[z]*Gamma[a, z], {z, n}] == (- 1)^(n)* Pochhammer[1 - a, n]*Exp[z]*Gamma[a - n, z]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [96 / 126]<div class="mw-collapsible-content"><code>{Plus[Complex[0.06772606154573046, -0.6693082179083164], Times[Complex[2.0864022336812553, 1.1398067350757155], DifferenceRoot[Function[{, } <- {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], Times[Power[Factorial[1], -1], Gamma[-1.5,</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>]] || <code>(x)^(1 - a)* exp(x)*GAMMA(a, x) <= 1</code> || <code>(x)^(1 - a)* Exp[x]*Gamma[a, x] <= 1</code> || 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>]] || <code>GAMMA(a)-GAMMA(a, x) >= ((x)^(a - 1))/(a)*(1 - exp(- x))</code> || <code>Gamma[a, 0, x] >= Divide[(x)^(a - 1),a]*(1 - Exp[- x])</code> || 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>]] || <code>(x)^(1 - a)* exp(x)*GAMMA(a, x) = 1 +(a - 1)/(x)*vartheta</code> || <code>(x)^(1 - a)* Exp[x]*Gamma[a, x] == 1 +Divide[a - 1,x]*\[CurlyTheta]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><code>180/180]: [[.8735840245+.8333333335*I <- {a = -1.5, x = 1.5, vartheta = 1/2*3^(1/2)+1/2*I}</code><br><code>-1.403124983+1.443375674*I <- {a = -1.5, x = 1.5, vartheta = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><code>{Complex[0.8735840235649492, 0.8333333333333331] <- {Rule[a, -1.5], Rule[x, 1.5], Rule[ϑ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-1.4031249827424481, 1.4433756729740643] <- {Rule[a, -1.5], Rule[x, 1.5], Rule[ϑ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/8.10.E4 8.10.E4] || [[Item:Q2557|<math>0 < \vartheta</math>]] || <code>0 < vartheta</code> || <code>0 < \[CurlyTheta]</code> || 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>]] || <code>A[n] < (x)^(1 - a)* exp(x)*GAMMA(a, x)</code> || <code>Subscript[A, n] < (x)^(1 - a)* Exp[x]*Gamma[a, x]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [75 / 300]<div class="mw-collapsible-content"><code>75/300]: [[1.5 < .4302083505 <- {a = -1.5, x = 1.5, A[n] = 1.5, n = 1}</code><br><code>1.5 < .4302083505 <- {a = -1.5, x = 1.5, A[n] = 1.5, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [195 / 300]<div class="mw-collapsible-content"><code>{Less[Complex[0.8660254037844387, 0.49999999999999994], 0.43020835059088497] <- {Rule[a, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[Subscript[A, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Less[Complex[0.8660254037844387, 0.49999999999999994], 0.43020835059088497] <- {Rule[a, -1.5], Rule[n, 2], Rule[x, 1.5], Rule[Subscript[A, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></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>]] || <code>(x)^(1 - a)* exp(x)*GAMMA(a, x) < B[n]</code> || <code>(x)^(1 - a)* Exp[x]*Gamma[a, x] < Subscript[B, n]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [105 / 300]<div class="mw-collapsible-content"><code>105/300]: [[.4302083505 < -1.5 <- {a = -1.5, x = 1.5, B[n] = -1.5, n = 1}</code><br><code>.4302083505 < -1.5 <- {a = -1.5, x = 1.5, B[n] = -1.5, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [225 / 300]<div class="mw-collapsible-content"><code>{Less[0.43020835059088497, Complex[0.8660254037844387, 0.49999999999999994]] <- {Rule[a, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[Subscript[B, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Less[0.43020835059088497, Complex[0.8660254037844387, 0.49999999999999994]] <- {Rule[a, -1.5], Rule[n, 2], Rule[x, 1.5], Rule[Subscript[B, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/8.10#Ex1 8.10#Ex1] || [[Item:Q2559|<math>A_{1} = \frac{x}{x+1-a}</math>]] || <code>A[1] = (x)/(x + 1 - a)</code> || <code>Subscript[A, 1] == Divide[x,x + 1 - a]</code> || 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>]] || <code>B[1] = (x + 1)/(x + 2 - a)</code> || <code>Subscript[B, 1] == Divide[x + 1,x + 2 - a]</code> || 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>]] || <code>A[2] = (x*(x + 3 - a))/((x)^(2)+ 2*(2 - a)* x +(1 - a)*(2 - a))</code> || <code>Subscript[A, 2] == Divide[x*(x + 3 - a),(x)^(2)+ 2*(2 - a)* x +(1 - a)*(2 - a)]</code> || 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>]] || <code>B[2] = ((x)^(2)+(5 - a)* x + 2)/((x)^(2)+ 2*(3 - a)* x +(2 - a)*(3 - a))</code> || <code>Subscript[B, 2] == Divide[(x)^(2)+(5 - a)* x + 2,(x)^(2)+ 2*(3 - a)* x +(2 - a)*(3 - a)]</code> || 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>]] || <code>I = int((t)^(a - 1)* exp(t), t = 0..x)</code> || <code>I == Integrate[(t)^(a - 1)* Exp[t], {t, 0, x}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [90 / 90]<div class="mw-collapsible-content"><code>90/90]: [[-2.374226751+.5000000000*I <- {I = 1/2*3^(1/2)+1/2*I, a = 1.5, x = 1.5}</code><br><code>.5451660792+.5000000000*I <- {I = 1/2*3^(1/2)+1/2*I, a = 1.5, x = .5}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [27 / 27]<div class="mw-collapsible-content"><code>{Complex[-2.240252154794788, -4.0113152157384396*^-17] <- {Rule[Complex[0, 1], 1], Rule[a, 1.5], Rule[x, 1.5]}</code><br><code>Complex[-1.240252154794788, -4.0113152157384396*^-17] <- {Rule[Complex[0, 1], 2], Rule[a, 1.5], Rule[x, 1.5]}</code><br></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>]] || <code>int((t)^(a - 1)* exp(t), t = 0..x) = GAMMA(a)*(x)^(a)* (- x)^(-(a))*(GAMMA(a)-GAMMA(a, - x))/GAMMA(a)</code> || <code>Error</code> || 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>]] || <code>((a + 1)*(a + 2)- x)/((a + 1)*(a + 2 + x)) < a*(x)^(- a)* exp(- x)*I</code> || <code>Divide[(a + 1)*(a + 2)- x,(a + 1)*(a + 2 + x)] < a*(x)^(- a)* Exp[- x]*I</code> || 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>]] || <code>c[a] = (GAMMA(1 + a))^(1/(a - 1))</code> || <code>Subscript[c, a] == (Gamma[1 + a])^(1/(a - 1))</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [39 / 40]<div class="mw-collapsible-content"><code>39/40]: [[-.9011204630+.5000000000*I <- {a = 1.5, c[a] = 1/2*3^(1/2)+1/2*I}</code><br><code>-2.267145867+.8660254040*I <- {a = 1.5, c[a] = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [39 / 40]<div class="mw-collapsible-content"><code>{Complex[-0.9011204638598199, 0.49999999999999994] <- {Rule[a, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-2.2671458676442584, 0.8660254037844387] <- {Rule[a, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/8.10#Ex6 8.10#Ex6] || [[Item:Q2566|<math>d_{a} = (\EulerGamma@{1+a})^{-1/a}</math>]] || <code>d[a] = (GAMMA(1 + a))^(- 1/ a)</code> || <code>Subscript[d, a] == (Gamma[1 + a])^(- 1/ a)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [40 / 40]<div class="mw-collapsible-content"><code>40/40]: [[.388914161e-1+.5000000000*I <- {a = 1.5, d[a] = 1/2*3^(1/2)+1/2*I}</code><br><code>-1.327133988+.8660254040*I <- {a = 1.5, d[a] = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [40 / 40]<div class="mw-collapsible-content"><code>{Complex[0.038891415918572037, 0.49999999999999994] <- {Rule[a, 1.5], Rule[Subscript[d, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-1.3271339878658663, 0.8660254037844387] <- {Rule[a, 1.5], Rule[Subscript[d, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| |-
| |
| | [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>]] || <code>(x)/(2*a)*((1 +(2)/(x))^(a)- 1) < (x)^(1 - a)* exp(x)*GAMMA(a, x)</code> || <code>Divide[x,2*a]*((1 +Divide[2,x])^(a)- 1) < (x)^(1 - a)* Exp[x]*Gamma[a, x]</code> || 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>]] || <code>(x)^(1 - a)* exp(x)*GAMMA(a, x) <= (x)/(a*c[a])*((1 +(c[a])/(x))^(a)- 1)</code> || <code>(x)^(1 - a)* Exp[x]*Gamma[a, x] <= Divide[x,a*Subscript[c, a]]*((1 +Divide[Subscript[c, a],x])^(a)- 1)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 30]<div class="mw-collapsible-content"><code>3/30]: [[.8100694499 <= .7912878480 <- {a = .5, x = 1.5, c[a] = 2}</code><br><code>.6556795425 <= .6180339885 <- {a = .5, x = .5, c[a] = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 30]<div class="mw-collapsible-content"><code>{LessEqual[0.8100694501969615, Complex[0.8808481919138387, -0.05137441674828974]] <- {Rule[a, 0.5], Rule[x, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>LessEqual[0.8100694501969615, Complex[1.0324243733930456, -0.1801654927820326]] <- {Rule[a, 0.5], Rule[x, 1.5], Rule[Subscript[c, a], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/8.10.E11 8.10.E11] || [[Item:Q2568|<math>(1-e^{-\alpha_{a}x})^{a} \leq \normincGammaP@{a}{x}</math>]] || <code>(1 - exp(- alpha[a]*x))^(a) <= (GAMMA(a)-GAMMA(a, x))/GAMMA(a)</code> || <code>(1 - Exp[- Subscript[\[Alpha], a]*x])^(a) <= GammaRegularized[a, 0, x]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [78 / 270]<div class="mw-collapsible-content"><code>78/270]: [[.8461432717 <= .6083748237 <- {a = 1.5, alpha = 1.5, x = 1.5, alpha[a] = 1.5}</code><br><code>.9262567903 <= .6083748237 <- {a = 1.5, alpha = 1.5, x = 1.5, alpha[a] = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [240 / 270]<div class="mw-collapsible-content"><code>{LessEqual[Complex[0.7016331692747775, 0.2500919864059583], 0.608374823728911] <- {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]]]}</code><br><code>LessEqual[Complex[-1.369574242346028, 2.679891945423719], 0.608374823728911] <- {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]]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/8.10.E11 8.10.E11] || [[Item:Q2568|<math>\normincGammaP@{a}{x} \leq (1-e^{-\beta_{a}x})^{a}</math>]] || <code>(GAMMA(a)-GAMMA(a, x))/GAMMA(a) <= (1 - exp(- beta[a]*x))^(a)</code> || <code>GammaRegularized[a, 0, x] <= (1 - Exp[- Subscript[\[Beta], a]*x])^(a)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 270]<div class="mw-collapsible-content"><code>30/270]: [[.6083748237 <= .3832643966 <- {a = 1.5, beta = 1.5, x = 1.5, beta[a] = .5}</code><br><code>.1987480431 <= .1040340193 <- {a = 1.5, beta = 1.5, x = .5, beta[a] = .5}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [192 / 270]<div class="mw-collapsible-content"><code>{LessEqual[0.608374823728911, Complex[0.7016331692747775, 0.2500919864059583]] <- {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]]]}</code><br><code>LessEqual[0.608374823728911, Complex[-1.369574242346028, 2.679891945423719]] <- {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]]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/8.10.E13 8.10.E13] || [[Item:Q2571|<math>\frac{\incGamma@{n}{n}}{\EulerGamma@{n}} < \frac{1}{2}</math>]] || <code>(GAMMA(n, n))/(GAMMA(n)) < (1)/(2)</code> || <code>Divide[Gamma[n, n],Gamma[n]] < Divide[1,2]</code> || 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>]] || <code>(1)/(2) < (GAMMA(n, n - 1))/(GAMMA(n))</code> || <code>Divide[1,2] < Divide[Gamma[n, n - 1],Gamma[n]]</code> || 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>]] || <code>GAMMA(a, z) = (z)^(a - 1)* exp(- z)*(sum((u[k])/((z)^(k)), k = 0..n - 1)+ R[n]*(a , z))</code> || <code>Gamma[a, z] == (z)^(a - 1)* Exp[- z]*(Sum[Divide[Subscript[u, k],(z)^(k)], {k, 0, n - 1}, GenerateConditions->None]+ Subscript[R, n]*(a , z))</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.1072320848e-1-.1480251451*I+(.9924715785e-1+.4087434498*I)*(1.000000000+0.*I+(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I)) <- {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}</code><br><code>.1072320848e-1-.1480251451*I+(.9924715785e-1+.4087434498*I)*(-1.165063509+1.250000000*I+(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I)) <- {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}</code><br></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>]] || <code>GAMMA(a)-GAMMA(a, z) = (z)^(a)* exp(- z)*sum(((z)^(k))/(pochhammer(a, k + 1)), k = 0..infinity)</code> || <code>Gamma[a, 0, z] == (z)^(a)* Exp[- z]*Sum[Divide[(z)^(k),Pochhammer[a, k + 1]], {k, 0, Infinity}, GenerateConditions->None]</code> || Successful || Successful || - || Successful [Tested: 7]
| |
| |-
| |
| | [https://dlmf.nist.gov/8.11#Ex1 8.11#Ex1] || [[Item:Q2579|<math>b_{0}(\lambda) = 1</math>]] || <code>b[0]*(lambda) = 1</code> || <code>Subscript[b, 0]*(\[Lambda]) == 1</code> || 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>]] || <code>b[1]*(lambda) = lambda</code> || <code>Subscript[b, 1]*(\[Lambda]) == \[Lambda]</code> || 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>]] || <code>b[2]*(lambda) = lambda*(2*lambda + 1)</code> || <code>Subscript[b, 2]*(\[Lambda]) == \[Lambda]*(2*\[Lambda]+ 1)</code> || 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>]] || <code>S[n]*(x) = (GAMMA(n + 1)-GAMMA(n + 1, n*x))/((n*x)^(n)* exp(- n*x))</code> || <code>Subscript[S, n]*(x) == Divide[Gamma[n + 1, 0, n*x],(n*x)^(n)* Exp[- n*x]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [90 / 90]<div class="mw-collapsible-content"><code>90/90]: [[-.22087941e-1+.7500000000*I <- {x = 1.5, S[n] = 1/2*3^(1/2)+1/2*I, n = 1}</code><br><code>-1.275525655+.7500000000*I <- {x = 1.5, S[n] = 1/2*3^(1/2)+1/2*I, n = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [90 / 90]<div class="mw-collapsible-content"><code>{Complex[-0.02208794121538471, 0.7499999999999999] <- {Rule[n, 1], Rule[x, 1.5], Rule[Subscript[S, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-1.2755256550317124, 0.7499999999999999] <- {Rule[n, 2], Rule[x, 1.5], Rule[Subscript[S, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></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>]] || <code>d[k]*(x) = ((- 1)^(k)* b[k]*(x))/((1 - x)^(2*k + 1))</code> || <code>Subscript[d, k]*(x) == Divide[(- 1)^(k)* Subscript[b, k]*(x),(1 - x)^(2*k + 1)]</code> || 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>]] || <code>lambda = z/ a</code> || <code>\[Lambda] == z/ a</code> || 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>]] || <code>eta = (2*(lambda - 1 - ln(lambda)))^(1/ 2)</code> || <code>\[Eta] == (2*(\[Lambda]- 1 - Log[\[Lambda]]))^(1/ 2)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><code>100/100]: [[.8206105237+1.019626504*I <- {eta = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I}</code><br><code>.2036159778+2.354396465*I <- {eta = 1/2*3^(1/2)+1/2*I, lambda = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><code>{Complex[0.8206105232686428, 1.019626504138681] <- {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[0.20361597732323333, 2.3543964646926674] <- {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></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>]] || <code>(1)/(2)*(eta)^(2) = lambda - 1 - ln(lambda)</code> || <code>Divide[1,2]*\[Eta]^(2) == \[Lambda]- 1 - Log[\[Lambda]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><code>100/100]: [[.3839745964+.4566114775*I <- {eta = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I}</code><br><code>1.750000000+1.661382400*I <- {eta = 1/2*3^(1/2)+1/2*I, lambda = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><code>{Complex[0.38397459621556135, 0.45661147749051817] <- {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[1.7499999999999998, 1.6613824005009756] <- {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/8.12#Ex4 8.12#Ex4] || [[Item:Q2596|<math>\deriv{\eta}{\lambda} = \frac{\lambda-1}{\lambda\eta}</math>]] || <code>diff(eta, lambda) = (lambda - 1)/(lambda*eta)</code> || <code>D[\[Eta], \[Lambda]] == Divide[\[Lambda]- 1,\[Lambda]*\[Eta]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><code>100/100]: [[-.3660254037-.3660254035*I <- {eta = 1/2*3^(1/2)+1/2*I, lambda = 1/2*3^(1/2)+1/2*I}</code><br><code>-1.732050807-.2266367838e-9*I <- {eta = 1/2*3^(1/2)+1/2*I, lambda = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [100 / 100]<div class="mw-collapsible-content"><code>{Complex[-0.3660254037844386, -0.3660254037844386] <- {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-1.7320508075688772, 3.3306690738754696*^-16] <- {Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></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>]] || <code>(GAMMA(a)-GAMMA(a, z))/GAMMA(a) = (1)/(2)*erfc(- eta*sqrt(a/ 2))- S*(a , eta)</code> || <code>GammaRegularized[a, 0, z] == Divide[1,2]*Erfc[- \[Eta]*Sqrt[a/ 2]]- S*(a , \[Eta])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.8724483635-.3325384943*I+(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I) <- {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}</code><br><code>.8436948583-.7685914925*I+(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I) <- {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)}</code><br></div></div> || Error
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| |-
| |
| | [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>]] || <code>GAMMA(a, z)/GAMMA(a) = (1)/(2)*erfc(eta*sqrt(a/ 2))+ S*(a , eta)</code> || <code>GammaRegularized[a, z] == Divide[1,2]*Erfc[\[Eta]*Sqrt[a/ 2]]+ S*(a , \[Eta])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-.8724483631+.3325384943*I-(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I) <- {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}</code><br><code>-.8436948582+.7685914925*I-(.8660254040+.5000000000*I)*(-1.5, .8660254040+.5000000000*I) <- {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)}</code><br></div></div> || Error
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| |-
| |
| | [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>]] || <code>(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)</code> || <code>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])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.1738836865-.4215091763*I+(-.5000000000+.8660254040*I)*(-1.5, .8660254040+.5000000000*I) <- {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}</code><br><code>-.5322485765+.1038051776*I+(-.5000000000+.8660254040*I)*(-1.5, .8660254040+.5000000000*I) <- {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)}</code><br></div></div> || Error
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| |-
| |
| | [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>]] || <code>(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)</code> || <code>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])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-.9809290254+.1461521622*I+(-.5000000000+.8660254040*I)*(-1.5, .8660254040+.5000000000*I) <- {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}</code><br><code>-.2747967621-.3791621909*I+(-.5000000000+.8660254040*I)*(-1.5, .8660254040+.5000000000*I) <- {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)}</code><br></div></div> || Error
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| |-
| |
| | [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>]] || <code>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)</code> || <code>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])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-.1738836865+.4215091762*I+(.5000000000-.8660254040*I)*(-1.5, .8660254040+.5000000000*I) <- {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}</code><br><code>.5322485766-.1038051776*I+(.5000000000-.8660254040*I)*(-1.5, .8660254040+.5000000000*I) <- {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)}</code><br></div></div> || Error
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| |-
| |
| | [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>]] || <code>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)</code> || <code>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])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.9809290254-.1461521621*I+(.5000000000-.8660254040*I)*(-1.5, .8660254040+.5000000000*I) <- {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}</code><br><code>.2747967620+.3791621907*I+(.5000000000-.8660254040*I)*(-1.5, .8660254040+.5000000000*I) <- {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)}</code><br></div></div> || Error
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| |-
| |
| | [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>]] || <code>(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))</code> || <code>Error</code> || Failure || Missing Macro Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.2923043261+1.961858052*I+(1.732050808+1.000000000*I)*(-1.5, .8660254040+.5000000000*I) <- {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}</code><br><code>-.7583243808+.5495935246*I+(1.732050808+1.000000000*I)*(-1.5, .8660254040+.5000000000*I) <- {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)}</code><br></div></div> || -
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| |-
| |
| | [https://dlmf.nist.gov/8.12#Ex6 8.12#Ex6] || [[Item:Q2604|<math>c_{0}(\eta) = \frac{1}{\mu}-\frac{1}{\eta}</math>]] || <code>c[0]*(eta) = (1)/(mu)-(1)/(eta)</code> || <code>Subscript[c, 0]*(\[Eta]) == Divide[1,\[Mu]]-Divide[1,\[Eta]]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
| |
| | [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>]] || <code>c[1]*(eta) = (1)/((eta)^(3))-(1)/((mu)^(3))-(1)/((mu)^(2))-(1)/(12*mu)</code> || <code>Subscript[c, 1]*(\[Eta]) == Divide[1,\[Eta]^(3)]-Divide[1,\[Mu]^(3)]-Divide[1,\[Mu]^(2)]-Divide[1,12*\[Mu]]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
| |
| | [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>]] || <code>c[k]*(eta) = (1)/(eta)*diff(c[k - 1]*(eta), eta)+(- 1)^(k)*(g[k])/(mu)</code> || <code>Subscript[c, k]*(\[Eta]) == Divide[1,\[Eta]]*D[Subscript[c, k - 1]*(\[Eta]), \[Eta]]+(- 1)^(k)*Divide[Subscript[g, k],\[Mu]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.5000000004+.8660254040*I <- {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}</code><br><code>-1.500000000+.8660254040*I <- {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}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[0.5000000000000001, 0.8660254037844386] <- {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]]]}</code><br><code>Complex[-1.5, 0.8660254037844386] <- {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]]]}</code><br></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>]] || <code>c[k]*(eta) = sum(d[k , n]*(eta)^(n), n = 0..infinity)</code> || <code>Subscript[c, k]*(\[Eta]) == Sum[Subscript[d, k , n]*\[Eta]^(n), {n, 0, Infinity}, GenerateConditions->None]</code> || 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>]] || <code>d[0 , n] = (n + 2)* alpha[n + 2]</code> || <code>Subscript[d, 0 , n] == (n + 2)* Subscript[\[Alpha], n + 2]</code> || 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>]] || <code>d[k , n] = (- 1)^(k)* g[k]*d[0 , n]+(n + 2)* d[k - 1 , n + 2]</code> || <code>Subscript[d, k , n] == (- 1)^(k)* Subscript[g, k]*Subscript[d, 0 , n]+(n + 2)* Subscript[d, k - 1 , n + 2]</code> || 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>]] || <code>lambda - 1 = eta +(1)/(3)*(eta)^(2)+ sum(alpha[n]*(eta)^(n), n = 3..infinity)</code> || <code>\[Lambda]- 1 == \[Eta]+Divide[1,3]*\[Eta]^(2)+ Sum[Subscript[\[Alpha], n]*\[Eta]^(n), {n, 3, Infinity}, GenerateConditions->None]</code> || 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>]] || <code>alpha[3] = (1)/(36)</code> || <code>Subscript[\[Alpha], 3] == Divide[1,36]</code> || 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>]] || <code>alpha[4] = -(1)/(270)</code> || <code>Subscript[\[Alpha], 4] == -Divide[1,270]</code> || 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>]] || <code>alpha[5] = (1)/(4320)</code> || <code>Subscript[\[Alpha], 5] == Divide[1,4320]</code> || 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>]] || <code>alpha[6] = (1)/(17010)</code> || <code>Subscript[\[Alpha], 6] == Divide[1,17010]</code> || 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>]] || <code>alpha[7] = -(139)/(5443200)</code> || <code>Subscript[\[Alpha], 7] == -Divide[139,5443200]</code> || 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>]] || <code>alpha[8] = (1)/(204120)</code> || <code>Subscript[\[Alpha], 8] == Divide[1,204120]</code> || 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>]] || <code>c[0]*(0) = -(1)/(3)</code> || <code>Subscript[c, 0]*(0) == -Divide[1,3]</code> || 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>]] || <code>c[1]*(0) = -(1)/(540)</code> || <code>Subscript[c, 1]*(0) == -Divide[1,540]</code> || 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>]] || <code>c[2]*(0) = (25)/(6048)</code> || <code>Subscript[c, 2]*(0) == Divide[25,6048]</code> || 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>]] || <code>c[3]*(0) = (101)/(155520)</code> || <code>Subscript[c, 3]*(0) == Divide[101,155520]</code> || 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>]] || <code>c[4]*(0) = -(3184811)/(3695155200)</code> || <code>Subscript[c, 4]*(0) == -Divide[3184811,3695155200]</code> || 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>]] || <code>c[5]*(0) = -(2745493)/(8151736320)</code> || <code>Subscript[c, 5]*(0) == -Divide[2745493,8151736320]</code> || 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>]] || <code>GAMMA(a, x)/GAMMA(a) = q</code> || <code>GammaRegularized[a, x] == q</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><code>180/180]: [[-.8512854784-.5000000000*I <- {a = -1.5, q = 1/2*3^(1/2)+1/2*I, x = 1.5}</code><br><code>-.5487148961-.5000000000*I <- {a = -1.5, q = 1/2*3^(1/2)+1/2*I, x = .5}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><code>{Complex[-0.8512854781447857, -0.49999999999999994] <- {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}</code><br><code>Complex[-0.5487148959215247, -0.49999999999999994] <- {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]}</code><br></div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/8.13.E1 8.13.E1] || [[Item:Q2633|<math>1+a^{-1} < x_{-}(a)</math>]] || <code>1 + (a)^(- 1) < x[-]*(a)</code> || <code>1 + (a)^(- 1) < Subscript[x, -]*(a)</code> || Error || Failure || - || Error
| |
| |-
| |
| | [https://dlmf.nist.gov/8.13.E1 8.13.E1] || [[Item:Q2633|<math>x_{-}(a) < \ln@@{|a|}</math>]] || <code>x[-]*(a) < ln(abs(a))</code> || <code>Subscript[x, -]*(a) < Log[Abs[a]]</code> || 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>]] || <code>int(exp(- a*x)*(GAMMA(b)-GAMMA(b, x))/(GAMMA(b)), x = 0..infinity) = ((1 + a)^(- b))/(a)</code> || <code>Integrate[Exp[- a*x]*Divide[Gamma[b, 0, x],Gamma[b]], {x, 0, Infinity}, GenerateConditions->None] == Divide[(1 + a)^(- b),a]</code> || 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>]] || <code>int(exp(- a*x)*GAMMA(b, x), x = 0..infinity) = GAMMA(b)*(1 -(1 + a)^(- b))/(a)</code> || <code>Integrate[Exp[- a*x]*Gamma[b, x], {x, 0, Infinity}, GenerateConditions->None] == Gamma[b]*Divide[1 -(1 + a)^(- b),a]</code> || 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>]] || <code>int((x)^(a - 1)* GAMMA(b)-GAMMA(b, x), x = 0..infinity) = -(GAMMA(a + b))/(a)</code> || <code>Integrate[(x)^(a - 1)* Gamma[b, 0, x], {x, 0, Infinity}, GenerateConditions->None] == -Divide[Gamma[a + b],a]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>3/3]: [[Float(infinity) <- {a = -1.5, b = 2}</code><br><code>Float(infinity) <- {a = -.5, b = 1.5}</code><br></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>]] || <code>int((x)^(a - 1)* GAMMA(b, x), x = 0..infinity) = (GAMMA(a + b))/(a)</code> || <code>Integrate[(x)^(a - 1)* Gamma[b, x], {x, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[a + b],a]</code> || 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>]] || <code>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))</code> || <code>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)]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [36 / 36]<div class="mw-collapsible-content"><code>36/36]: [[Float(infinity) <- {a = -1.5, b = 2, s = 1.5}</code><br><code>Float(infinity) <- {a = -1.5, b = 2, s = .5}</code><br></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>]] || <code>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))</code> || <code>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)]</code> || 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>]] || <code>GAMMA(a)-GAMMA(a, lambda*x) = (lambda)^(a)* sum(GAMMA(a + k)-GAMMA(a + k, x)*((1 - lambda)^(k))/(factorial(k)), k = 0..infinity)</code> || <code>Gamma[a, 0, \[Lambda]*x] == \[Lambda]^(a)* Sum[Gamma[a + k, 0, x]*Divide[(1 - \[Lambda])^(k),(k)!], {k, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [90 / 90]<div class="mw-collapsible-content"><code>90/90]: [[Float(infinity)+Float(infinity)*I <- {a = 1.5, lambda = 1/2*3^(1/2)+1/2*I, x = 1.5}</code><br><code>Float(infinity)+Float(infinity)*I <- {a = 1.5, lambda = 1/2*3^(1/2)+1/2*I, x = .5}</code><br></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>]] || <code>int(t^(a-1)*(1-t)^(b-1), t = 0 .. x) = int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..x)</code> || <code>Beta[x, a, b] == Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, x}, GenerateConditions->None]</code> || 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>]] || <code>Error</code> || <code>BetaRegularized[x, a, b] == Beta[x, a, b]/ Beta[a, b]</code> || Missing Macro Error || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [45 / 108]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 1.5]}</code><br><code>Indeterminate <- {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 0.5]}</code><br></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>]] || <code>Beta(a, b) = (GAMMA(a)*GAMMA(b))/(GAMMA(a + b))</code> || <code>Beta[a, b] == Divide[Gamma[a]*Gamma[b],Gamma[a + b]]</code> || 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>]] || <code>Error</code> || <code>BetaRegularized[x, a, b] == 1 - BetaRegularized[1 - x, b, a]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [39 / 108]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 1.5]}</code><br><code>Indeterminate <- {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 0.5]}</code><br></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>]] || <code>Error</code> || <code>BetaRegularized[x, m, n - m + 1] == Sum[Binomial[n,j]*(x)^(j)*(1 - x)^(n - j), {j, m, n}, GenerateConditions->None]</code> || 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>]] || <code>Error</code> || <code>BetaRegularized[x, a, a] == Divide[1,2]*BetaRegularized[4*x*(1 - x), a, Divide[1,2]]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 6]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[a, -1.5], Rule[x, 0.5]}</code><br><code>Indeterminate <- {Rule[a, -0.5], Rule[x, 0.5]}</code><br></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>]] || <code>int(t^(a-1)*(1-t)^(b-1), t = 0 .. x) = ((x)^(a))/(a)*hypergeom([a, 1 - b], [a + 1], x)</code> || <code>Beta[x, a, b] == Divide[(x)^(a),a]*Hypergeometric2F1[a, 1 - b, a + 1, x]</code> || Failure || Successful || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 108]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[a, -2], Rule[b, -1.5], Rule[x, 1.5]}</code><br><code>Indeterminate <- {Rule[a, -2], Rule[b, -1.5], Rule[x, 0.5]}</code><br></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>]] || <code>int(t^(a-1)*(1-t)^(b-1), t = 0 .. x) = ((x)^(a)*(1 - x)^(b))/(a)*hypergeom([a + b, 1], [a + 1], x)</code> || <code>Beta[x, a, b] == Divide[(x)^(a)*(1 - x)^(b),a]*Hypergeometric2F1[a + b, 1, a + 1, x]</code> || Failure || Successful || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [15 / 108]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[a, -2], Rule[b, -1.5], Rule[x, 1.5]}</code><br><code>Indeterminate <- {Rule[a, -2], Rule[b, -1.5], Rule[x, 0.5]}</code><br></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>]] || <code>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))</code> || <code>Beta[x, a, b] == Divide[(x)^(a)*(1 - x)^(b - 1),a]*Hypergeometric2F1[1, 1 - b, a + 1, Divide[x,x - 1]]</code> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [46 / 108]<div class="mw-collapsible-content"><code>{Complex[4.9960036108132044*^-15, -27.48893571891068] <- {Rule[a, -1.5], Rule[b, -2], Rule[x, 1.5]}</code><br><code>Complex[3.191891195797325*^-15, -27.48893571891068] <- {Rule[a, -1.5], Rule[b, -2], Rule[x, 2]}</code><br></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>]] || <code>Error</code> || <code>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]</code> || Missing Macro Error || Aborted || - || Skipped - Because timed out
| |
| |-
| |
| | [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>]] || <code>Error</code> || <code>(a + b)* BetaRegularized[x, a, b] == a*BetaRegularized[x, a + 1, b]+ b*BetaRegularized[x, a, b + 1]</code> || Missing Macro Error || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [33 / 108]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 1.5]}</code><br><code>Indeterminate <- {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 0.5]}</code><br></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>]] || <code>Error</code> || <code>(a + b*x)* BetaRegularized[x, a, b] == x*b*BetaRegularized[x, a - 1, b + 1]+ a*BetaRegularized[x, a + 1, b]</code> || Missing Macro Error || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [36 / 108]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 1.5]}</code><br><code>Indeterminate <- {Rule[a, -1.5], Rule[b, -1.5], Rule[x, 0.5]}</code><br></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>]] || <code>Error</code> || <code>a*BetaRegularized[x, a + 1, b] == (a + c*x)* BetaRegularized[x, a, b]- c*x*BetaRegularized[x, a - 1, b]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [246 / 300]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[x, 1.5]}</code><br><code>Indeterminate <- {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[x, 0.5]}</code><br></div></div>
| |
| |-
| |
| | [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>]] || <code>d[2*m] = (m*(b - m)* x)/((a + 2*m - 1)*(a + 2*m))</code> || <code>Subscript[d, 2*m] == Divide[m*(b - m)* x,(a + 2*m - 1)*(a + 2*m)]</code> || 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>]] || <code>d[2*m + 1] = -((a + m)*(a + b + m)* x)/((a + 2*m)*(a + 2*m + 1))</code> || <code>Subscript[d, 2*m + 1] == -Divide[(a + m)*(a + b + m)* x,(a + 2*m)*(a + 2*m + 1)]</code> || 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>]] || <code>Error</code> || <code>BetaRegularized[x, m, n] == (1 - x)^(n)* Sum[Binomial[n + j - 1,j]*(x)^(j), {j, m, Infinity}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || Successful [Tested: 9]
| |
| |-
| |
| | [https://dlmf.nist.gov/8.18.E2 8.18.E2] || [[Item:Q2668|<math>\xi = -\ln@@{x}</math>]] || <code>xi = - ln(x)</code> || <code>\[Xi] == - Log[x]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><code>30/30]: [[1.271490512+.5000000000*I <- {x = 1.5, xi = 1/2*3^(1/2)+1/2*I}</code><br><code>-.945348919e-1+.8660254040*I <- {x = 1.5, xi = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><code>{Complex[1.271490511892603, 0.49999999999999994] <- {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-0.0945348918918354, 0.8660254037844387] <- {Rule[x, 1.5], Rule[ξ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></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>]] || <code>a*F[k + 1] = (k + b - a*xi)* F[k]+ k*xi*F[k - 1]</code> || <code>a*Subscript[F, k + 1] == (k + b - a*\[Xi])* Subscript[F, k]+ k*\[Xi]*Subscript[F, k - 1]</code> || 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>]] || <code>F[0] = (a)^(- b)* GAMMA(b, a*xi)/GAMMA(b)</code> || <code>Subscript[F, 0] == (a)^(- b)* GammaRegularized[b, a*\[Xi]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[1.253924788+1.407498490*I <- {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}</code><br><code>-.1121006157+1.773523894*I <- {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)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[1.2539247882576399, 1.4074984905445393] <- {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]]]}</code><br><code>Complex[-0.11210061552679867, 1.7735238943289782] <- {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]]]}</code><br></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>]] || <code>F[1] = (b - a*xi)/(a)*F[0]+((xi)^(b)* exp(- a*xi))/(a*GAMMA(b))</code> || <code>Subscript[F, 1] == Divide[b - a*\[Xi],a]*Subscript[F, 0]+Divide[\[Xi]^(b)* Exp[- a*\[Xi]],a*Gamma[b]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[2.329643864+4.621882749*I <- {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}</code><br><code>.9636184598+4.987908153*I <- {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)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[2.32964386182885, 4.621882746395113] <- {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]]]}</code><br><code>Complex[0.9636184580444114, 4.9879081501795515] <- {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]]]}</code><br></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>]] || <code>((1 - exp(- t))/(t))^(b - 1) = sum(d[k]*(t - xi)^(k), k = 0..infinity)</code> || <code>(Divide[1 - Exp[- t],t])^(b - 1) == Sum[Subscript[d, k]*(t - \[Xi])^(k), {k, 0, Infinity}, GenerateConditions->None]</code> || 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>]] || <code>d[0] = ((1 - x)/(xi))^(b - 1)</code> || <code>Subscript[d, 0] == (Divide[1 - x,\[Xi]])^(b - 1)</code> || 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>]] || <code>d[1] = (x*xi + x - 1)/((1 - x)* xi)*(b - 1)* d[0]</code> || <code>Subscript[d, 1] == Divide[x*\[Xi]+ x - 1,(1 - x)* \[Xi]]*(b - 1)* Subscript[d, 0]</code> || 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>]] || <code>x[0] = a/(a + b)</code> || <code>Subscript[x, 0] == a/(a + b)</code> || 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>]] || <code>-(1)/(2)*(eta)^(2) = x[0]*ln((x)/(x[0]))+(1 - x[0])* ln((1 - x)/(1 - x[0]))</code> || <code>-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]]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.580000474e-1+.458917392e-1*I <- {eta = 1/2*3^(1/2)+1/2*I, x = 1.5, x[0] = 1/2*3^(1/2)+1/2*I}</code><br><code>2.269862383+1.019641337*I <- {eta = 1/2*3^(1/2)+1/2*I, x = 1.5, x[0] = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[0.058000047924774145, 0.04589173995258988] <- {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]]]}</code><br><code>Complex[2.2698623824536366, 1.0196413375539057] <- {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]]]}</code><br></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>]] || <code>c[0]*(eta) = (1)/(eta)-(sqrt(x[0]*(1 - x[0])))/(x - x[0])</code> || <code>Subscript[c, 0]*(\[Eta]) == Divide[1,\[Eta]]-Divide[Sqrt[Subscript[x, 0]*(1 - Subscript[x, 0])],x - Subscript[x, 0]]</code> || 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>]] || <code>c[0]*(0) = (1 - 2*x[0])/(3*sqrt(x[0]*(1 - x[0])))</code> || <code>Subscript[c, 0]*(0) == Divide[1 - 2*Subscript[x, 0],3*Sqrt[Subscript[x, 0]*(1 - Subscript[x, 0])]]</code> || 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>]] || <code>mu*ln(zeta)- zeta = ln(x)+ mu*ln(1 - x)+(1 + mu)* ln(1 + mu)- mu</code> || <code>\[Mu]*Log[\[Zeta]]- \[Zeta] == Log[x]+ \[Mu]*Log[1 - x]+(1 + \[Mu])* Log[1 + \[Mu]]- \[Mu]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [299 / 300]<div class="mw-collapsible-content"><code>299/300]: [[.405976146-2.738439399*I <- {mu = 1/2*3^(1/2)+1/2*I, x = 1.5, zeta = 1/2*3^(1/2)+1/2*I}</code><br><code>.9866033870-1.744115280*I <- {mu = 1/2*3^(1/2)+1/2*I, x = 1.5, zeta = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [299 / 300]<div class="mw-collapsible-content"><code>{Complex[0.4059761460255107, -2.7384393975724306] <- {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]]]}</code><br><code>Complex[1.0560847852373059, 1.7517066341083583] <- {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]]]}</code><br></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>]] || <code>h[0]*(zeta , mu) = mu*((1)/(zeta - mu)-((1 + mu)^(- 3/ 2))/(x[0]- x))</code> || <code>Subscript[h, 0]*(\[Zeta], \[Mu]) == \[Mu]*(Divide[1,\[Zeta]- \[Mu]]-Divide[(1 + \[Mu])^(- 3/ 2),Subscript[x, 0]- x])</code> || 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>]] || <code>h[0]*(mu , mu) = (1)/(3)*((1 - mu)/(sqrt(1 + mu))- 1)</code> || <code>Subscript[h, 0]*(\[Mu], \[Mu]) == Divide[1,3]*(Divide[1 - \[Mu],Sqrt[1 + \[Mu]]]- 1)</code> || 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>]] || <code>Error</code> || <code>BetaRegularized[x, a, b] == p</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [105 / 108]<div class="mw-collapsible-content"><code>{DirectedInfinity[] <- {Rule[a, -1.5], Rule[b, -1.5], Rule[p, 0.5], Rule[x, 1.5]}</code><br><code>Indeterminate <- {Rule[a, -1.5], Rule[b, -1.5], Rule[p, 0.5], Rule[x, 0.5]}</code><br></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>]] || <code>Ei(p, z) = (z)^(p - 1)* GAMMA(1 - p, z)</code> || <code>ExpIntegralE[p, z] == (z)^(p - 1)* Gamma[1 - p, z]</code> || 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>]] || <code>Ei(p, z) = (z)^(p - 1)* int((exp(- t))/((t)^(p)), t = z..infinity)</code> || <code>ExpIntegralE[p, z] == (z)^(p - 1)* Integrate[Divide[Exp[- t],(t)^(p)], {t, z, Infinity}, GenerateConditions->None]</code> || 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>]] || <code>Ei(p, z) = int((exp(- z*t))/((t)^(p)), t = 1..infinity)</code> || <code>ExpIntegralE[p, z] == Integrate[Divide[Exp[- z*t],(t)^(p)], {t, 1, Infinity}, GenerateConditions->None]</code> || 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>]] || <code>Ei(p, z) = ((z)^(p - 1)* exp(- z))/(GAMMA(p))*int(((t)^(p - 1)* exp(- z*t))/(1 + t), t = 0..infinity)</code> || <code>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]</code> || 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>]] || <code>Ei(0, z) = (z)^(- 1)* exp(- z)</code> || <code>ExpIntegralE[0, z] == (z)^(- 1)* Exp[- z]</code> || 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>]] || <code>Ei(p, 0) = (1)/(p - 1)</code> || <code>ExpIntegralE[p, 0] == Divide[1,p - 1]</code> || 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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><code>21/21]: [[-1.393548628-1.498247032*I <- {z = 1/2*3^(1/2)+1/2*I, n = 1, n = 2}</code><br><code>.4577249979+1.994294304*I <- {z = 1/2*3^(1/2)+1/2*I, n = 2, n = 2}</code><br></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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || Successful [Tested: 21] || <div class="toccolours mw-collapsible mw-collapsed">Failed [16 / 21]<div class="mw-collapsible-content"><code>{Plus[Complex[0.3691288000469654, -0.2016559825387078], Times[Complex[0.2188469268397846, -0.35920360372711485], Plus[-1.0, DifferenceRoot[Function[{, } <- {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]]]}</code><br><code>Plus[Complex[-0.0256870546657635, -0.10579058942927923], Times[Complex[0.1094234634198923, -0.17960180186355743], Plus[-2.0, DifferenceRoot[Function[{, } <- {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, Rational[1, 6]], Pi]]]}</code><br></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>]] || <code>Ei(p, z) = (z)^(p - 1)* GAMMA(1 - p)- sum(((- z)^(k))/(factorial(k)*(1 - p + k)), k = 0..infinity)</code> || <code>ExpIntegralE[p, z] == (z)^(p - 1)* Gamma[1 - p]- Sum[Divide[(- z)^(k),(k)!*(1 - p + k)], {k, 0, Infinity}, GenerateConditions->None]</code> || 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>]] || <code>Ei(p, z) = GAMMA(1 - p)*((z)^(p - 1)- exp(- z)*sum(((z)^(k))/(GAMMA(2 - p + k)), k = 0..infinity))</code> || <code>ExpIntegralE[p, z] == Gamma[1 - p]*((z)^(p - 1)- Exp[- z]*Sum[Divide[(z)^(k),Gamma[2 - p + k]], {k, 0, Infinity}, GenerateConditions->None])</code> || 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>]] || <code>p*Ei(p + 1, z)+ z*Ei(p, z) = exp(- z)</code> || <code>p*ExpIntegralE[p + 1, z]+ z*ExpIntegralE[p, z] == Exp[- z]</code> || 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>]] || <code>diff(Ei(p, z), z) = - Ei(p - 1, z)</code> || <code>D[ExpIntegralE[p, z], z] == - ExpIntegralE[p - 1, z]</code> || 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>]] || <code>diff(exp(z)*Ei(p, z), z) = exp(z)*Ei(p, z)*(1 +(p - 1)/(z))-(1)/(z)</code> || <code>D[Exp[z]*ExpIntegralE[p, z], z] == Exp[z]*ExpIntegralE[p, z]*(1 +Divide[p - 1,z])-Divide[1,z]</code> || 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>]] || <code>diff(Ei(p, z), [p$(j)]) = (- 1)^(j)* int((ln(t))^(j)* (t)^(- p)* exp(- z*t), t = 1..infinity)</code> || <code>D[ExpIntegralE[p, z], {p, j}] == (- 1)^(j)* Integrate[(Log[t])^(j)* (t)^(- p)* Exp[- z*t], {t, 1, Infinity}, GenerateConditions->None]</code> || 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>]] || <code>Ei(p, z) = (z)^(p - 1)* exp(- z)*KummerU(p, p, z)</code> || <code>ExpIntegralE[p, z] == (z)^(p - 1)* Exp[- z]*HypergeometricU[p, p, z]</code> || 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>]] || <code>(n - 1)/(n)*Ei(n, x) < Ei(n + 1, x)</code> || <code>Divide[n - 1,n]*ExpIntegralE[n, x] < ExpIntegralE[n + 1, x]</code> || 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>]] || <code>Ei(n + 1, x) < Ei(n, x)</code> || <code>ExpIntegralE[n + 1, x] < ExpIntegralE[n, x]</code> || 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>]] || <code>(Ei(n, x))^(2) < Ei(n - 1, x)*Ei(n + 1, x)</code> || <code>(ExpIntegralE[n, x])^(2) < ExpIntegralE[n - 1, x]*ExpIntegralE[n + 1, x]</code> || 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>]] || <code>(1)/(x + n) < exp(x)*Ei(n, x)</code> || <code>Divide[1,x + n] < Exp[x]*ExpIntegralE[n, x]</code> || 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>]] || <code>exp(x)*Ei(n, x) <= (1)/(x + n - 1)</code> || <code>Exp[x]*ExpIntegralE[n, x] <= Divide[1,x + n - 1]</code> || 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>]] || <code>diff((Ei(n, x))/(Ei(n - 1, x)), x) > 0</code> || <code>D[Divide[ExpIntegralE[n, x],ExpIntegralE[n - 1, x]], x] > 0</code> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| |-
| |
| | [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>]] || <code>int(Ei(p - 1, t), t = z..infinity) = Ei(p, z)</code> || <code>Integrate[ExpIntegralE[p - 1, t], {t, z, Infinity}, GenerateConditions->None] == ExpIntegralE[p, z]</code> || 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>]] || <code>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))</code> || <code>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])</code> || 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>]] || <code>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))</code> || <code>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)]</code> || Failure || Aborted || Skipped - Because timed out || Successful [Tested: 64]
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| |-
| |
| | [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>]] || <code>int(Ei(p, t)*Ei(q, t), t = 0..infinity) = (L*(p)+ L*(q))/(p + q - 1)</code> || <code>Integrate[ExpIntegralE[p, t]*ExpIntegralE[q, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[L*(p)+ L*(q),p + q - 1]</code> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [80 / 80]<div class="mw-collapsible-content"><code>{Complex[-0.8698344324715543, -0.7499999999999999] <- {Rule[L, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[p, 1.5], Rule[q, 1.5]}</code><br><code>Complex[0.26794919243112303, -0.9999999999999999] <- {Rule[L, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[p, 1.5], Rule[q, 0.5]}</code><br></div></div>
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| |-
| |
| | [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>]] || <code>L*(p) = int(exp(- t)*Ei(p, t), t = 0..infinity)</code> || <code>L*(p) == Integrate[Exp[- t]*ExpIntegralE[p, t], {t, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><code>{Complex[0.8698344324715546, 0.7499999999999999] <- {Rule[L, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[p, 1.5]}</code><br><code>Complex[-1.1377836249026771, 0.24999999999999997] <- {Rule[L, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[p, 0.5]}</code><br></div></div>
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| |
| | [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>]] || <code>int(exp(- t)*Ei(p, t), t = 0..infinity) = (1)/(2*p)*hypergeom([1, 1], [1 + p], (1)/(2))</code> || <code>Integrate[Exp[- t]*ExpIntegralE[p, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2*p]*Hypergeometric2F1[1, 1, 1 + p, Divide[1,2]]</code> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
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| |-
| |
| | [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>]] || <code>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))</code> || <code>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])</code> || Failure || Successful || Manual Skip! || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 70]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[n, 3], Rule[p, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Indeterminate <- {Rule[n, 3], Rule[p, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [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>]] || <code>A[k + 1]*(lambda) = (1 - 2*k*lambda)* A[k]*(lambda)+ lambda*(lambda + 1)* diff(A[k]*(lambda), lambda)</code> || <code>Subscript[A, k + 1]*(\[Lambda]) == (1 - 2*k*\[Lambda])* Subscript[A, k]*(\[Lambda])+ \[Lambda]*(\[Lambda]+ 1)* D[Subscript[A, k]*(\[Lambda]), \[Lambda]]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-.5000000000+.133974596*I <- {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}</code><br><code>-.4999999993+2.133974597*I <- {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}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[-0.49999999999999944, 4.133974596215561] <- {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]]]}</code><br><code>Complex[-1.8660254037844384, 3.7679491924311224] <- {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]]]}</code><br></div></div>
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| |
| | [https://dlmf.nist.gov/8.20#Ex1 8.20#Ex1] || [[Item:Q2718|<math>A_{1}(\lambda) = 1</math>]] || <code>A[1]*(lambda) = 1</code> || <code>Subscript[A, 1]*(\[Lambda]) == 1</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
| |
| | [https://dlmf.nist.gov/8.20#Ex2 8.20#Ex2] || [[Item:Q2719|<math>A_{2}(\lambda) = 1-2\lambda</math>]] || <code>A[2]*(lambda) = 1 - 2*lambda</code> || <code>Subscript[A, 2]*(\[Lambda]) == 1 - 2*\[Lambda]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
| |
| | [https://dlmf.nist.gov/8.20#Ex3 8.20#Ex3] || [[Item:Q2720|<math>A_{3}(\lambda) = 1-8\lambda+6\lambda^{2}</math>]] || <code>A[3]*(lambda) = 1 - 8*lambda + 6*(lambda)^(2)</code> || <code>Subscript[A, 3]*(\[Lambda]) == 1 - 8*\[Lambda]+ 6*\[Lambda]^(2)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
| |
| | [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>]] || <code>int((t)^(a - 1)* exp(+ I*t), t = 0..infinity) = exp(+(1)/(2)*Pi*I*a)*GAMMA(a)</code> || <code>Integrate[(t)^(a - 1)* Exp[+ I*t], {t, 0, Infinity}, GenerateConditions->None] == Exp[+Divide[1,2]*Pi*I*a]*Gamma[a]</code> || 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>]] || <code>int((t)^(a - 1)* exp(- I*t), t = 0..infinity) = exp(-(1)/(2)*Pi*I*a)*GAMMA(a)</code> || <code>Integrate[(t)^(a - 1)* Exp[- I*t], {t, 0, Infinity}, GenerateConditions->None] == Exp[-Divide[1,2]*Pi*I*a]*Gamma[a]</code> || 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>]] || <code>(GAMMA(p))/(2*Pi)*(z)^(1 - p)* Ei(p, z) = (GAMMA(p))/(2*Pi)*GAMMA(1 - p, z)</code> || <code>Divide[Gamma[p],2*Pi]*(z)^(1 - p)* ExpIntegralE[p, z] == Divide[Gamma[p],2*Pi]*Gamma[1 - p, z]</code> || Successful || Successful || - || Successful [Tested: 35]
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| |-
| |
| | [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>]] || <code>zeta[x]*(s) = sum((k)^(- s)* (GAMMA(s)-GAMMA(s, k*x))/GAMMA(s), k = 1..infinity)</code> || <code>Subscript[\[Zeta], x]*(s) == Sum[(k)^(- s)* GammaRegularized[s, 0, k*x], {k, 1, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || Manual Skip! || Skipped - Because timed out
| |
| |-
| |
| |}
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