Results of Gamma Function: Difference between revisions

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{| class="wikitable sortable" style="margin: 0;"
; Notation : [[5.1|5.1 Special Notation]]<br>
|-
; Properties : [[5.2|5.2 Definitions]]<br>[[5.3|5.3 Graphics]]<br>[[5.4|5.4 Special Values and Extrema]]<br>[[5.5|5.5 Functional Relations]]<br>[[5.6|5.6 Inequalities]]<br>[[5.7|5.7 Series Expansions]]<br>[[5.8|5.8 Infinite Products]]<br>[[5.9|5.9 Integral Representations]]<br>[[5.10|5.10 Continued Fractions]]<br>[[5.11|5.11 Asymptotic Expansions]]<br>[[5.12|5.12 Beta Function]]<br>[[5.13|5.13 Integrals]]<br>[[5.14|5.14 Multidimensional Integrals]]<br>[[5.15|5.15 Polygamma Functions]]<br>[[5.16|5.16 Sums]]<br>[[5.17|5.17 Barnes’ <math>\BarnesG</math> -Function (Double Gamma Function)]]<br>[[5.18|5.18 <math>q</math> -Gamma and <math>q</math> -Beta Functions]]<br>
! scope="col" style="position: sticky; top: 0;" | DLMF
; Applications : [[5.19|5.19 Mathematical Applications]]<br>[[5.20|5.20 Physical Applications]]<br>
! scope="col" style="position: sticky; top: 0;" | Formula
; Computation : [[5.21|5.21 Methods of Computation]]<br>[[5.22|5.22 Tables]]<br>[[5.23|5.23 Approximations]]<br>[[5.24|5.24 Software]]<br>
! scope="col" style="position: sticky; top: 0;" | Constraints
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
|-
|-
| [https://dlmf.nist.gov/5.2.E1 5.2.E1] || [[Item:Q2026|<math>\EulerGamma@{z} = \int_{0}^{\infty}e^{-t}t^{z-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z} = \int_{0}^{\infty}e^{-t}t^{z-1}\diff{t}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z) = int(exp(- t)*(t)^(z - 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z] == Integrate[Exp[- t]*(t)^(z - 1), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 5]
|-
| [https://dlmf.nist.gov/5.2.E2 5.2.E2] || [[Item:Q2027|<math>\digamma@{z} = \EulerGamma'@{z}/\EulerGamma@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z} = \EulerGamma'@{z}/\EulerGamma@{z}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>Psi(z) = diff( GAMMA(z), z$(1) )/GAMMA(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z] == D[Gamma[z], {z, 1}]/Gamma[z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.2#Ex1 5.2#Ex1] || [[Item:Q2029|<math>\Pochhammersym{a}{0} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Pochhammersym{a}{0} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>pochhammer(a, 0) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Pochhammer[a, 0] == 1</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 6]
|-
| [https://dlmf.nist.gov/5.2.E5 5.2.E5] || [[Item:Q2031|<math>\Pochhammersym{a}{n} = \EulerGamma@{a+n}/\EulerGamma@{a}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Pochhammersym{a}{n} = \EulerGamma@{a+n}/\EulerGamma@{a}</syntaxhighlight> || <math>\realpart@@{(a+n)} > 0, \realpart@@{a} > 0</math> || <syntaxhighlight lang=mathematica>pochhammer(a, n) = GAMMA(a + n)/GAMMA(a)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Pochhammer[a, n] == Gamma[a + n]/Gamma[a]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.2.E6 5.2.E6] || [[Item:Q2032|<math>\Pochhammersym{-a}{n} = (-1)^{n}\Pochhammersym{a-n+1}{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Pochhammersym{-a}{n} = (-1)^{n}\Pochhammersym{a-n+1}{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>pochhammer(- a, n) = (- 1)^(n)* pochhammer(a - n + 1, n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Pochhammer[- a, n] == (- 1)^(n)* Pochhammer[a - n + 1, n]</syntaxhighlight> || Failure || Failure || Successful [Tested: 18] || Successful [Tested: 18]
|-
| [https://dlmf.nist.gov/5.2#Ex3 5.2#Ex3] || [[Item:Q2034|<math>\Pochhammersym{a}{2n} = 2^{2n}\Pochhammersym{\frac{a}{2}}{n}\Pochhammersym{\frac{a+1}{2}}{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Pochhammersym{a}{2n} = 2^{2n}\Pochhammersym{\frac{a}{2}}{n}\Pochhammersym{\frac{a+1}{2}}{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>pochhammer(a, 2*n) = (2)^(2*n)* pochhammer((a)/(2), n)*pochhammer((a + 1)/(2), n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Pochhammer[a, 2*n] == (2)^(2*n)* Pochhammer[Divide[a,2], n]*Pochhammer[Divide[a + 1,2], n]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 18]
|-
| [https://dlmf.nist.gov/5.2#Ex4 5.2#Ex4] || [[Item:Q2035|<math>\Pochhammersym{a}{2n+1} = 2^{2n+1}\Pochhammersym{\frac{a}{2}}{n+1}\Pochhammersym{\frac{a+1}{2}}{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Pochhammersym{a}{2n+1} = 2^{2n+1}\Pochhammersym{\frac{a}{2}}{n+1}\Pochhammersym{\frac{a+1}{2}}{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>pochhammer(a, 2*n + 1) = (2)^(2*n + 1)* pochhammer((a)/(2), n + 1)*pochhammer((a + 1)/(2), n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Pochhammer[a, 2*n + 1] == (2)^(2*n + 1)* Pochhammer[Divide[a,2], n + 1]*Pochhammer[Divide[a + 1,2], n]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 18]
|-
| [https://dlmf.nist.gov/5.4#Ex1 5.4#Ex1] || [[Item:Q2036|<math>\EulerGamma@{1} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{1} = 1</syntaxhighlight> || <math>\realpart@@{1} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(1) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[1] == 1</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.4#Ex2 5.4#Ex2] || [[Item:Q2037|<math>n! = \EulerGamma@{n+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>n! = \EulerGamma@{n+1}</syntaxhighlight> || <math>\realpart@@{(n+1)} > 0</math> || <syntaxhighlight lang=mathematica>factorial(n) = GAMMA(n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(n)! == Gamma[n + 1]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.4.E3 5.4.E3] || [[Item:Q2039|<math>|\EulerGamma@{iy}| = \left(\frac{\pi}{y\sinh@{\pi y}}\right)^{1/2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\EulerGamma@{iy}| = \left(\frac{\pi}{y\sinh@{\pi y}}\right)^{1/2}</syntaxhighlight> || <math>\realpart@@{(\iunit y)} > 0</math> || <syntaxhighlight lang=mathematica>abs(GAMMA(I*y)) = ((Pi)/(y*sinh(Pi*y)))^(1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Gamma[I*y]] == (Divide[Pi,y*Sinh[Pi*y]])^(1/2)</syntaxhighlight> || Failure || Failure || Error || Skip - No test values generated
|-
| [https://dlmf.nist.gov/5.4.E4 5.4.E4] || [[Item:Q2040|<math>\EulerGamma@{\tfrac{1}{2}+\iunit y}\EulerGamma@{\tfrac{1}{2}-\iunit y} = \left|\EulerGamma@{\tfrac{1}{2}+\iunit y}\right|^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{\tfrac{1}{2}+\iunit y}\EulerGamma@{\tfrac{1}{2}-\iunit y} = \left|\EulerGamma@{\tfrac{1}{2}+\iunit y}\right|^{2}</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+\iunit y)} > 0, \realpart@@{(\tfrac{1}{2}-\iunit y)} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA((1)/(2)+ I*y)*GAMMA((1)/(2)- I*y) = (abs(GAMMA((1)/(2)+ I*y)))^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[Divide[1,2]+ I*y]*Gamma[Divide[1,2]- I*y] == (Abs[Gamma[Divide[1,2]+ I*y]])^(2)</syntaxhighlight> || Failure || Failure || Successful [Tested: 6] || Successful [Tested: 6]
|-
| [https://dlmf.nist.gov/5.4.E4 5.4.E4] || [[Item:Q2040|<math>\left|\EulerGamma@{\tfrac{1}{2}+\iunit y}\right|^{2} = \frac{\pi}{\cosh@{\pi y}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left|\EulerGamma@{\tfrac{1}{2}+\iunit y}\right|^{2} = \frac{\pi}{\cosh@{\pi y}}</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+\iunit y)} > 0, \realpart@@{(\tfrac{1}{2}-\iunit y)} > 0</math> || <syntaxhighlight lang=mathematica>(abs(GAMMA((1)/(2)+ I*y)))^(2) = (Pi)/(cosh(Pi*y))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Abs[Gamma[Divide[1,2]+ I*y]])^(2) == Divide[Pi,Cosh[Pi*y]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 6] || Successful [Tested: 6]
|-
| [https://dlmf.nist.gov/5.4.E5 5.4.E5] || [[Item:Q2041|<math>\EulerGamma@{\tfrac{1}{4}+\iunit y}\EulerGamma@{\tfrac{3}{4}-\iunit y} = \frac{\pi\sqrt{2}}{\cosh@{\pi y}+\iunit\sinh@{\pi y}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{\tfrac{1}{4}+\iunit y}\EulerGamma@{\tfrac{3}{4}-\iunit y} = \frac{\pi\sqrt{2}}{\cosh@{\pi y}+\iunit\sinh@{\pi y}}</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{4}+\iunit y)} > 0, \realpart@@{(\tfrac{3}{4}-\iunit y)} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA((1)/(4)+ I*y)*GAMMA((3)/(4)- I*y) = (Pi*sqrt(2))/(cosh(Pi*y)+ I*sinh(Pi*y))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[Divide[1,4]+ I*y]*Gamma[Divide[3,4]- I*y] == Divide[Pi*Sqrt[2],Cosh[Pi*y]+ I*Sinh[Pi*y]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 6]
|-
| [https://dlmf.nist.gov/5.4.E6 5.4.E6] || [[Item:Q2042|<math>\EulerGamma@{\tfrac{1}{2}} = \pi^{1/2}\\</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{\tfrac{1}{2}} = \pi^{1/2}\\</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2})} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA((1)/(2)) = (Pi)^(1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[Divide[1,2]] == (Pi)^(1/2)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.4.E6 5.4.E6] || [[Item:Q2042|<math>\pi^{1/2}\\ = 1.77245\;38509\;05516\;02729\;\dots</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\pi^{1/2}\\ = 1.77245\;38509\;05516\;02729\;\dots</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2})} > 0</math> || <syntaxhighlight lang=mathematica>(Pi)^(1/2) = 1.77245385090551602729</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Pi)^(1/2) == 1.77245385090551602729</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.4.E7 5.4.E7] || [[Item:Q2043|<math>\EulerGamma@{\tfrac{1}{3}} = 2.67893\;85347\;07747\;63365\;\dots</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{\tfrac{1}{3}} = 2.67893\;85347\;07747\;63365\;\dots</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{3})} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA((1)/(3)) = 2.67893853470774763365</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[Divide[1,3]] == 2.67893853470774763365</syntaxhighlight> || Failure || Successful || Successful [Tested: 0] || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.4.E8 5.4.E8] || [[Item:Q2044|<math>\EulerGamma@{\tfrac{2}{3}} = 1.35411\;79394\;26400\;41694\;\dots</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{\tfrac{2}{3}} = 1.35411\;79394\;26400\;41694\;\dots</syntaxhighlight> || <math>\realpart@@{(\tfrac{2}{3})} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA((2)/(3)) = 1.35411793942640041694</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[Divide[2,3]] == 1.35411793942640041694</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.4.E9 5.4.E9] || [[Item:Q2045|<math>\EulerGamma@{\tfrac{1}{4}} = 3.62560\;99082\;21908\;31193\;\dots</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{\tfrac{1}{4}} = 3.62560\;99082\;21908\;31193\;\dots</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{4})} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA((1)/(4)) = 3.62560990822190831193</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[Divide[1,4]] == 3.62560990822190831193</syntaxhighlight> || Failure || Successful || Successful [Tested: 0] || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.4.E10 5.4.E10] || [[Item:Q2046|<math>\EulerGamma@{\tfrac{3}{4}} = 1.22541\;67024\;65177\;64512\;\dots</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{\tfrac{3}{4}} = 1.22541\;67024\;65177\;64512\;\dots</syntaxhighlight> || <math>\realpart@@{(\tfrac{3}{4})} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA((3)/(4)) = 1.22541670246517764512</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[Divide[3,4]] == 1.22541670246517764512</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.4.E11 5.4.E11] || [[Item:Q2047|<math>\EulerGamma'@{1} = -\EulerConstant</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma'@{1} = -\EulerConstant</syntaxhighlight> || <math>\realpart@@{1} > 0</math> || <syntaxhighlight lang=mathematica>subs( temp=1, diff( GAMMA(temp), temp$(1) ) ) = - gamma</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Gamma[temp], {temp, 1}]/.temp-> 1) == - EulerGamma</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.4#Ex3 5.4#Ex3] || [[Item:Q2048|<math>\digamma@{1} = -\EulerConstant</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{1} = -\EulerConstant</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(1) = - gamma</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[1] == - EulerGamma</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.4#Ex4 5.4#Ex4] || [[Item:Q2049|<math>\digamma'@{1} = \tfrac{1}{6}\pi^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma'@{1} = \tfrac{1}{6}\pi^{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>subs( temp=1, diff( Psi(temp), temp$(1) ) ) = (1)/(6)*(Pi)^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[PolyGamma[temp], {temp, 1}]/.temp-> 1) == Divide[1,6]*(Pi)^(2)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.4#Ex5 5.4#Ex5] || [[Item:Q2050|<math>\digamma@{\tfrac{1}{2}} = -\EulerConstant-2\ln@@{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{\tfrac{1}{2}} = -\EulerConstant-2\ln@@{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi((1)/(2)) = - gamma - 2*ln(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[Divide[1,2]] == - EulerGamma - 2*Log[2]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.4#Ex6 5.4#Ex6] || [[Item:Q2051|<math>\digamma'@{\tfrac{1}{2}} = \tfrac{1}{2}\pi^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma'@{\tfrac{1}{2}} = \tfrac{1}{2}\pi^{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>subs( temp=(1)/(2), diff( Psi(temp), temp$(1) ) ) = (1)/(2)*(Pi)^(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[PolyGamma[temp], {temp, 1}]/.temp-> Divide[1,2]) == Divide[1,2]*(Pi)^(2)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.4.E14 5.4.E14] || [[Item:Q2052|<math>\digamma@{n+1} = \sum_{k=1}^{n}\frac{1}{k}-\EulerConstant</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{n+1} = \sum_{k=1}^{n}\frac{1}{k}-\EulerConstant</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(n + 1) = sum((1)/(k), k = 1..n)- gamma</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[n + 1] == Sum[Divide[1,k], {k, 1, n}, GenerateConditions->None]- EulerGamma</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.4.E16 5.4.E16] || [[Item:Q2054|<math>\imagpart@@{\digamma@{iy}} = \frac{1}{2y}+\frac{\pi}{2}\coth@{\pi y}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\imagpart@@{\digamma@{iy}} = \frac{1}{2y}+\frac{\pi}{2}\coth@{\pi y}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Im(Psi(I*y)) = (1)/(2*y)+(Pi)/(2)*coth(Pi*y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Im[PolyGamma[I*y]] == Divide[1,2*y]+Divide[Pi,2]*Coth[Pi*y]</syntaxhighlight> || Failure || Failure || Successful [Tested: 6] || Successful [Tested: 6]
|-
| [https://dlmf.nist.gov/5.4.E17 5.4.E17] || [[Item:Q2055|<math>\imagpart@@{\digamma@{\tfrac{1}{2}+iy}} = \frac{\pi}{2}\tanh@{\pi y}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\imagpart@@{\digamma@{\tfrac{1}{2}+iy}} = \frac{\pi}{2}\tanh@{\pi y}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Im(Psi((1)/(2)+ I*y)) = (Pi)/(2)*tanh(Pi*y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Im[PolyGamma[Divide[1,2]+ I*y]] == Divide[Pi,2]*Tanh[Pi*y]</syntaxhighlight> || Failure || Failure || Successful [Tested: 6] || Successful [Tested: 6]
|-
| [https://dlmf.nist.gov/5.4.E18 5.4.E18] || [[Item:Q2056|<math>\imagpart@@{\digamma@{1+iy}} = -\frac{1}{2y}+\frac{\pi}{2}\coth@{\pi y}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\imagpart@@{\digamma@{1+iy}} = -\frac{1}{2y}+\frac{\pi}{2}\coth@{\pi y}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Im(Psi(1 + I*y)) = -(1)/(2*y)+(Pi)/(2)*coth(Pi*y)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Im[PolyGamma[1 + I*y]] == -Divide[1,2*y]+Divide[Pi,2]*Coth[Pi*y]</syntaxhighlight> || Failure || Failure || Successful [Tested: 6] || Successful [Tested: 6]
|-
| [https://dlmf.nist.gov/5.4.E19 5.4.E19] || [[Item:Q2057|<math>\digamma@{\frac{p}{q}} = -\EulerConstant-\ln@@{q}-\frac{\pi}{2}\cot@{\frac{\pi p}{q}}+\frac{1}{2}\sum_{k=1}^{q-1}\cos@{\frac{2\pi kp}{q}}\ln@{2-2\cos@{\frac{2\pi k}{q}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{\frac{p}{q}} = -\EulerConstant-\ln@@{q}-\frac{\pi}{2}\cot@{\frac{\pi p}{q}}+\frac{1}{2}\sum_{k=1}^{q-1}\cos@{\frac{2\pi kp}{q}}\ln@{2-2\cos@{\frac{2\pi k}{q}}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi((p)/(q)) = - gamma - ln(q)-(Pi)/(2)*cot((Pi*p)/(q))+(1)/(2)*sum(cos((2*Pi*k*p)/(q))*ln(2 - 2*cos((2*Pi*k)/(q))), k = 1..q - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[Divide[p,q]] == - EulerGamma - Log[q]-Divide[Pi,2]*Cot[Divide[Pi*p,q]]+Divide[1,2]*Sum[Cos[Divide[2*Pi*k*p,q]]*Log[2 - 2*Cos[Divide[2*Pi*k,q]]], {k, 1, q - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/5.5.E1 5.5.E1] || [[Item:Q2059|<math>\EulerGamma@{z+1} = z\EulerGamma@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z+1} = z\EulerGamma@{z}</syntaxhighlight> || <math>\realpart@@{(z+1)} > 0, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z + 1) = z*GAMMA(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z + 1] == z*Gamma[z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 5]
|-
| [https://dlmf.nist.gov/5.5.E2 5.5.E2] || [[Item:Q2060|<math>\digamma@{z+1} = \digamma@{z}+\frac{1}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z+1} = \digamma@{z}+\frac{1}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z + 1) = Psi(z)+(1)/(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z + 1] == PolyGamma[z]+Divide[1,z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/5.5.E3 5.5.E3] || [[Item:Q2061|<math>\EulerGamma@{z}\EulerGamma@{1-z} = \pi/\sin@{\pi z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z}\EulerGamma@{1-z} = \pi/\sin@{\pi z}</syntaxhighlight> || <math>\realpart@@{z} > 0, \realpart@@{(1-z)} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z)*GAMMA(1 - z) = Pi/sin(Pi*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z]*Gamma[1 - z] == Pi/Sin[Pi*z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.5.E4 5.5.E4] || [[Item:Q2062|<math>\digamma@{z}-\digamma@{1-z} = -\pi/\tan@{\pi z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z}-\digamma@{1-z} = -\pi/\tan@{\pi z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z)- Psi(1 - z) = - Pi/tan(Pi*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z]- PolyGamma[1 - z] == - Pi/Tan[Pi*z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.5.E5 5.5.E5] || [[Item:Q2063|<math>\EulerGamma@{2z} = \pi^{-1/2}2^{2z-1}\EulerGamma@{z}\EulerGamma@{z+\tfrac{1}{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{2z} = \pi^{-1/2}2^{2z-1}\EulerGamma@{z}\EulerGamma@{z+\tfrac{1}{2}}</syntaxhighlight> || <math>\realpart@@{(2z)} > 0, \realpart@@{z} > 0, \realpart@@{(z+\tfrac{1}{2})} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(2*z) = (Pi)^(- 1/2)* (2)^(2*z - 1)* GAMMA(z)*GAMMA(z +(1)/(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[2*z] == (Pi)^(- 1/2)* (2)^(2*z - 1)* Gamma[z]*Gamma[z +Divide[1,2]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 5]
|-
| [https://dlmf.nist.gov/5.5.E6 5.5.E6] || [[Item:Q2064|<math>\EulerGamma@{nz} = (2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\EulerGamma@{z+\frac{k}{n}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{nz} = (2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\EulerGamma@{z+\frac{k}{n}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(n*z) = (2*Pi)^((1 - n)/2)* (n)^(n*z -(1/2))* product(GAMMA(z +(k)/(n)), k = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[n*z] == (2*Pi)^((1 - n)/2)* (n)^(n*z -(1/2))* Product[Gamma[z +Divide[k,n]], {k, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 15] || Successful [Tested: 15]
|-
| [https://dlmf.nist.gov/5.5.E7 5.5.E7] || [[Item:Q2065|<math>\prod_{k=1}^{n-1}\EulerGamma@{\frac{k}{n}} = (2\pi)^{(n-1)/2}n^{-1/2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\prod_{k=1}^{n-1}\EulerGamma@{\frac{k}{n}} = (2\pi)^{(n-1)/2}n^{-1/2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>product(GAMMA((k)/(n)), k = 1..n - 1) = (2*Pi)^((n - 1)/2)* (n)^(- 1/2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Product[Gamma[Divide[k,n]], {k, 1, n - 1}, GenerateConditions->None] == (2*Pi)^((n - 1)/2)* (n)^(- 1/2)</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[n, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[n, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.5.E8 5.5.E8] || [[Item:Q2066|<math>\digamma@{2z} = \tfrac{1}{2}\left(\digamma@{z}+\digamma@{z+\tfrac{1}{2}}\right)+\ln@@{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{2z} = \tfrac{1}{2}\left(\digamma@{z}+\digamma@{z+\tfrac{1}{2}}\right)+\ln@@{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(2*z) = (1)/(2)*(Psi(z)+ Psi(z +(1)/(2)))+ ln(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[2*z] == Divide[1,2]*(PolyGamma[z]+ PolyGamma[z +Divide[1,2]])+ Log[2]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/5.5.E9 5.5.E9] || [[Item:Q2067|<math>\digamma@{nz} = \frac{1}{n}\sum_{k=0}^{n-1}\digamma@{z+\frac{k}{n}}+\ln@@{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{nz} = \frac{1}{n}\sum_{k=0}^{n-1}\digamma@{z+\frac{k}{n}}+\ln@@{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(n*z) = (1)/(n)*sum(Psi(z +(k)/(n)), k = 0..n - 1)+ ln(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[n*z] == Divide[1,n]*Sum[PolyGamma[z +Divide[k,n]], {k, 0, n - 1}, GenerateConditions->None]+ Log[n]</syntaxhighlight> || Failure || Successful || Successful [Tested: 21] || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/5.6.E1 5.6.E1] || [[Item:Q2068|<math>1 < (2\pi)^{-1/2}x^{(1/2)-x}e^{x}\EulerGamma@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>1 < (2\pi)^{-1/2}x^{(1/2)-x}e^{x}\EulerGamma@{x}</syntaxhighlight> || <math>\realpart@@{x} > 0</math> || <syntaxhighlight lang=mathematica>1 < (2*Pi)^(- 1/2)* (x)^((1/2)- x)* exp(x)*GAMMA(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>1 < (2*Pi)^(- 1/2)* (x)^((1/2)- x)* Exp[x]*Gamma[x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.6.E1 5.6.E1] || [[Item:Q2068|<math>(2\pi)^{-1/2}x^{(1/2)-x}e^{x}\EulerGamma@{x} < e^{1/(12x)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(2\pi)^{-1/2}x^{(1/2)-x}e^{x}\EulerGamma@{x} < e^{1/(12x)}</syntaxhighlight> || <math>\realpart@@{x} > 0</math> || <syntaxhighlight lang=mathematica>(2*Pi)^(- 1/2)* (x)^((1/2)- x)* exp(x)*GAMMA(x) < exp(1/(12*x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(2*Pi)^(- 1/2)* (x)^((1/2)- x)* Exp[x]*Gamma[x] < Exp[1/(12*x)]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.6.E2 5.6.E2] || [[Item:Q2069|<math>\frac{1}{\EulerGamma@{x}}+\frac{1}{\EulerGamma@{1/x}} \leq 2</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{x}}+\frac{1}{\EulerGamma@{1/x}} \leq 2</syntaxhighlight> || <math>\realpart@@{x} > 0, \realpart@@{(1/x)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(x))+(1)/(GAMMA(1/x)) <= 2</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[x]]+Divide[1,Gamma[1/x]] <= 2</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.6.E3 5.6.E3] || [[Item:Q2070|<math>\frac{1}{(\EulerGamma@{x})^{2}}+\frac{1}{(\EulerGamma@{1/x})^{2}} \leq 2</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{(\EulerGamma@{x})^{2}}+\frac{1}{(\EulerGamma@{1/x})^{2}} \leq 2</syntaxhighlight> || <math>\realpart@@{x} > 0, \realpart@@{(1/x)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/((GAMMA(x))^(2))+(1)/((GAMMA(1/x))^(2)) <= 2</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,(Gamma[x])^(2)]+Divide[1,(Gamma[1/x])^(2)] <= 2</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.6.E4 5.6.E4] || [[Item:Q2071|<math>\frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}} < (x+1)^{1-s}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}} < (x+1)^{1-s}</syntaxhighlight> || <math>0 < s, s < 1, \realpart@@{(x+1)} > 0, \realpart@@{(x+s)} > 0</math> || <syntaxhighlight lang=mathematica>(GAMMA(x + 1))/(GAMMA(x + s)) < (x + 1)^(1 - s)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Gamma[x + 1],Gamma[x + s]] < (x + 1)^(1 - s)</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.6.E5 5.6.E5] || [[Item:Q2072|<math>\exp@{(1-s)\digamma@{x+s^{1/2}}} \leq \frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\exp@{(1-s)\digamma@{x+s^{1/2}}} \leq \frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}}</syntaxhighlight> || <math>0 < s, s < 1, \realpart@@{(x+1)} > 0, \realpart@@{(x+s)} > 0</math> || <syntaxhighlight lang=mathematica>exp((1 - s)*Psi(x + (s)^(1/2))) <= (GAMMA(x + 1))/(GAMMA(x + s))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[(1 - s)*PolyGamma[x + (s)^(1/2)]] <= Divide[Gamma[x + 1],Gamma[x + s]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.6.E5 5.6.E5] || [[Item:Q2072|<math>\frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}} \leq \exp@{(1-s)\digamma@{x+\tfrac{1}{2}(s+1)}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}} \leq \exp@{(1-s)\digamma@{x+\tfrac{1}{2}(s+1)}}</syntaxhighlight> || <math>0 < s, s < 1, \realpart@@{(x+1)} > 0, \realpart@@{(x+s)} > 0</math> || <syntaxhighlight lang=mathematica>(GAMMA(x + 1))/(GAMMA(x + s)) <= exp((1 - s)*Psi(x +(1)/(2)*(s + 1)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[Gamma[x + 1],Gamma[x + s]] <= Exp[(1 - s)*PolyGamma[x +Divide[1,2]*(s + 1)]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.6.E6 5.6.E6] || [[Item:Q2073|<math>|\EulerGamma@{x+\iunit y}| \leq |\EulerGamma@{x}|</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\EulerGamma@{x+\iunit y}| \leq |\EulerGamma@{x}|</syntaxhighlight> || <math>\realpart@@{(x+\iunit y)} > 0, \realpart@@{x} > 0</math> || <syntaxhighlight lang=mathematica>abs(GAMMA(x + I*y)) <= abs(GAMMA(x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Gamma[x + I*y]] <= Abs[Gamma[x]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 18] || Successful [Tested: 18]
|-
| [https://dlmf.nist.gov/5.6.E7 5.6.E7] || [[Item:Q2074|<math>|\EulerGamma@{x+\iunit y}| \geq (\sech@{\pi y})^{1/2}\EulerGamma@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\EulerGamma@{x+\iunit y}| \geq (\sech@{\pi y})^{1/2}\EulerGamma@{x}</syntaxhighlight> || <math>x \geq \tfrac{1}{2}, \realpart@@{(x+\iunit y)} > 0, \realpart@@{x} > 0</math> || <syntaxhighlight lang=mathematica>abs(GAMMA(x + I*y)) >= (sech(Pi*y))^(1/2)* GAMMA(x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Gamma[x + I*y]] >= (Sech[Pi*y])^(1/2)* Gamma[x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 18] || Successful [Tested: 18]
|-
| [https://dlmf.nist.gov/5.6.E8 5.6.E8] || [[Item:Q2075|<math>\left|\frac{\EulerGamma@{z+a}}{\EulerGamma@{z+b}}\right| \leq \frac{1}{|z|^{b-a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left|\frac{\EulerGamma@{z+a}}{\EulerGamma@{z+b}}\right| \leq \frac{1}{|z|^{b-a}}</syntaxhighlight> || <math>\realpart@@{(z+a)} > 0, \realpart@@{(z+b)} > 0</math> || <syntaxhighlight lang=mathematica>abs((GAMMA(z + a))/(GAMMA(z + b))) <= (1)/((abs(z))^(b - a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Divide[Gamma[z + a],Gamma[z + b]]] <= Divide[1,(Abs[z])^(b - a)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 83]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .5333333334 <= .1250000000
Test Values: {a = -1.5, b = 1.5, z = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.000000000 <= .5000000000
Test Values: {a = -1.5, b = -.5, z = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.333333334 <= .2500000000
Test Values: {a = -1.5, b = .5, z = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2954089752 <= .8838834764e-1
Test Values: {a = -1.5, b = 2, z = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [35 / 95]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: False
Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, 2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: False
Test Values: {Rule[a, -1.5], Rule[b, -0.5], Rule[z, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.6.E9 5.6.E9] || [[Item:Q2076|<math>|\EulerGamma@{z}| \leq (2\pi)^{1/2}|z|^{x-(1/2)}e^{-\pi|y|/2}\exp@{\tfrac{1}{6}|z|^{-1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>|\EulerGamma@{z}| \leq (2\pi)^{1/2}|z|^{x-(1/2)}e^{-\pi|y|/2}\exp@{\tfrac{1}{6}|z|^{-1}}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>abs(GAMMA(x + y*I)) <= (2*Pi)^(1/2)*(abs(x + y*I))^(x -(1/2))* exp(- Pi*abs(y)/2)*exp((1)/(6)*(abs(x + y*I))^(- 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Abs[Gamma[x + y*I]] <= (2*Pi)^(1/2)*(Abs[x + y*I])^(x -(1/2))* Exp[- Pi*Abs[y]/2]*Exp[Divide[1,6]*(Abs[x + y*I])^(- 1)]</syntaxhighlight> || Failure || Failure || Successful [Tested: 18] || Successful [Tested: 18]
|-
| [https://dlmf.nist.gov/5.7.E1 5.7.E1] || [[Item:Q2077|<math>\frac{1}{\EulerGamma@{z}} = \sum_{k=1}^{\infty}c_{k}z^{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{z}} = \sum_{k=1}^{\infty}c_{k}z^{k}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(z)) = sum(c[k]*(z)^(k), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[z]] == Sum[Subscript[c, k]*(z)^(k), {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [50 / 50]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.444337041-.9752791869*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, c[k] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.444337041+1.756771621*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, c[k] = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.287713767-.9752791869*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, c[k] = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.287713767+1.756771621*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, c[k] = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [50 / 50]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.0783116366515544, 0.3907462172966202], Times[-1.0, NSum[Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[1, k]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.0783116366515544, 0.3907462172966202], Times[-1.0, NSum[Times[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], k]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[c, k], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.7.E3 5.7.E3] || [[Item:Q2079|<math>\ln@@{\EulerGamma@{1+z}} = -\ln@{1+z}+z(1-\EulerConstant)+\sum_{k=2}^{\infty}(-1)^{k}(\Riemannzeta@{k}-1)\frac{z^{k}}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\ln@@{\EulerGamma@{1+z}} = -\ln@{1+z}+z(1-\EulerConstant)+\sum_{k=2}^{\infty}(-1)^{k}(\Riemannzeta@{k}-1)\frac{z^{k}}{k}</syntaxhighlight> || <math>|z| < 2, \realpart@@{(1+z)} > 0</math> || <syntaxhighlight lang=mathematica>ln(GAMMA(1 + z)) = - ln(1 + z)+ z*(1 - gamma)+ sum((- 1)^(k)*(Zeta(k)- 1)*((z)^(k))/(k), k = 2..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Gamma[1 + z]] == - Log[1 + z]+ z*(1 - EulerGamma)+ Sum[(- 1)^(k)*(Zeta[k]- 1)*Divide[(z)^(k),k], {k, 2, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 6] || Successful [Tested: 6]
|-
| [https://dlmf.nist.gov/5.7.E4 5.7.E4] || [[Item:Q2080|<math>\digamma@{1+z} = -\EulerConstant+\sum_{k=2}^{\infty}(-1)^{k}\Riemannzeta@{k}z^{k-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{1+z} = -\EulerConstant+\sum_{k=2}^{\infty}(-1)^{k}\Riemannzeta@{k}z^{k-1}</syntaxhighlight> || <math>|z| < 1</math> || <syntaxhighlight lang=mathematica>Psi(1 + z) = - gamma + sum((- 1)^(k)* Zeta(k)*(z)^(k - 1), k = 2..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[1 + z] == - EulerGamma + Sum[(- 1)^(k)* Zeta[k]*(z)^(k - 1), {k, 2, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 1] || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.7.E5 5.7.E5] || [[Item:Q2081|<math>\digamma@{1+z} = \frac{1}{2z}-\frac{\pi}{2}\cot@{\pi z}+\frac{1}{z^{2}-1}+1-\EulerConstant-\sum_{k=1}^{\infty}(\Riemannzeta@{2k+1}-1)z^{2k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{1+z} = \frac{1}{2z}-\frac{\pi}{2}\cot@{\pi z}+\frac{1}{z^{2}-1}+1-\EulerConstant-\sum_{k=1}^{\infty}(\Riemannzeta@{2k+1}-1)z^{2k}</syntaxhighlight> || <math>|z| < 2, z \neq 0</math> || <syntaxhighlight lang=mathematica>Psi(1 + z) = (1)/(2*z)-(Pi)/(2)*cot(Pi*z)+(1)/((z)^(2)- 1)+ 1 - gamma - sum((Zeta(2*k + 1)- 1)*(z)^(2*k), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[1 + z] == Divide[1,2*z]-Divide[Pi,2]*Cot[Pi*z]+Divide[1,(z)^(2)- 1]+ 1 - EulerGamma - Sum[(Zeta[2*k + 1]- 1)*(z)^(2*k), {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Error || Successful || - || Successful [Tested: 6]
|-
| [https://dlmf.nist.gov/5.7.E6 5.7.E6] || [[Item:Q2082|<math>\digamma@{z} = -\EulerConstant-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(k+z)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z} = -\EulerConstant-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(k+z)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z) = - gamma -(1)/(z)+ sum((z)/(k*(k + z)), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z] == - EulerGamma -Divide[1,z]+ Sum[Divide[z,k*(k + z)], {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/5.7.E6 5.7.E6] || [[Item:Q2082|<math>-\EulerConstant-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(k+z)} = -\EulerConstant+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+z}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\EulerConstant-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(k+z)} = -\EulerConstant+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+z}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>- gamma -(1)/(z)+ sum((z)/(k*(k + z)), k = 1..infinity) = - gamma + sum((1)/(k + 1)-(1)/(k + z), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- EulerGamma -Divide[1,z]+ Sum[Divide[z,k*(k + z)], {k, 1, Infinity}, GenerateConditions->None] == - EulerGamma + Sum[Divide[1,k + 1]-Divide[1,k + z], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 7] || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/5.7.E7 5.7.E7] || [[Item:Q2083|<math>\digamma@{\frac{z+1}{2}}-\digamma@{\frac{z}{2}} = 2\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k+z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{\frac{z+1}{2}}-\digamma@{\frac{z}{2}} = 2\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k+z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi((z + 1)/(2))- Psi((z)/(2)) = 2*sum(((- 1)^(k))/(k + z), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[Divide[z + 1,2]]- PolyGamma[Divide[z,2]] == 2*Sum[Divide[(- 1)^(k),k + z], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/5.7.E8 5.7.E8] || [[Item:Q2084|<math>\imagpart@@{\digamma@{1+\iunit y}} = \sum_{k=1}^{\infty}\frac{y}{k^{2}+y^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\imagpart@@{\digamma@{1+\iunit y}} = \sum_{k=1}^{\infty}\frac{y}{k^{2}+y^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Im(Psi(1 + I*y)) = sum((y)/((k)^(2)+ (y)^(2)), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Im[PolyGamma[1 + I*y]] == Sum[Divide[y,(k)^(2)+ (y)^(2)], {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 6] || Successful [Tested: 6]
|-
| [https://dlmf.nist.gov/5.8.E2 5.8.E2] || [[Item:Q2086|<math>\frac{1}{\EulerGamma@{z}} = ze^{\EulerConstant z}\prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)e^{-z/k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{z}} = ze^{\EulerConstant z}\prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)e^{-z/k}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(z)) = z*exp(gamma*z)*product((1 +(z)/(k))*exp(- z/k), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[z]] == z*Exp[EulerGamma*z]*Product[(1 +Divide[z,k])*Exp[- z/k], {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 5]
|-
| [https://dlmf.nist.gov/5.8.E3 5.8.E3] || [[Item:Q2087|<math>\left|\frac{\EulerGamma@{x}}{\EulerGamma@{x+\iunit y}}\right|^{2} = \prod_{k=0}^{\infty}\left(1+\frac{y^{2}}{(x+k)^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left|\frac{\EulerGamma@{x}}{\EulerGamma@{x+\iunit y}}\right|^{2} = \prod_{k=0}^{\infty}\left(1+\frac{y^{2}}{(x+k)^{2}}\right)</syntaxhighlight> || <math>\realpart@@{x} > 0, \realpart@@{(x+\iunit y)} > 0</math> || <syntaxhighlight lang=mathematica>(abs((GAMMA(x))/(GAMMA(x + I*y))))^(2) = product(1 +((y)^(2))/((x + k)^(2)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Abs[Divide[Gamma[x],Gamma[x + I*y]]])^(2) == Product[1 +Divide[(y)^(2),(x + k)^(2)], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -6.243891895+0.*I
Test Values: {x = 1.5, y = -1.5, x = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -6.243891895+0.*I
Test Values: {x = 1.5, y = 1.5, x = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.210463144+0.*I
Test Values: {x = 1.5, y = -.5, x = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.210463144+0.*I
Test Values: {x = 1.5, y = .5, x = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 18]
|-
| [https://dlmf.nist.gov/5.9.E1 5.9.E1] || [[Item:Q2090|<math>\frac{1}{\mu}\EulerGamma@{\frac{\nu}{\mu}}\frac{1}{z^{\nu/\mu}} = \int_{0}^{\infty}\exp@{-zt^{\mu}}t^{\nu-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\mu}\EulerGamma@{\frac{\nu}{\mu}}\frac{1}{z^{\nu/\mu}} = \int_{0}^{\infty}\exp@{-zt^{\mu}}t^{\nu-1}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/(mu)*GAMMA((nu)/(mu))*(1)/((z)^(nu/mu)) = int(exp(- z*(t)^(mu))*(t)^(nu - 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,\[Mu]]*Gamma[Divide[\[Nu],\[Mu]]]*Divide[1,(z)^(\[Nu]/\[Mu])] == Integrate[Exp[- z*(t)^\[Mu]]*(t)^(\[Nu]- 1), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Successful [Tested: 300]
|-
| [https://dlmf.nist.gov/5.9.E2 5.9.E2] || [[Item:Q2091|<math>\frac{1}{\EulerGamma@{z}} = \frac{1}{2\pi i}\int_{-\infty}^{(0+)}e^{t}t^{-z}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{z}} = \frac{1}{2\pi i}\int_{-\infty}^{(0+)}e^{t}t^{-z}\diff{t}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(z)) = (1)/(2*Pi*I)*int(exp(t)*(t)^(- z), t = - infinity..(0 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[z]] == Divide[1,2*Pi*I]*Integrate[Exp[t]*(t)^(- z), {t, - Infinity, (0 +)}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
|-
| [https://dlmf.nist.gov/5.9.E3 5.9.E3] || [[Item:Q2092|<math>c^{-z}\EulerGamma@{z} = \int_{-\infty}^{\infty}|t|^{2z-1}e^{-ct^{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>c^{-z}\EulerGamma@{z} = \int_{-\infty}^{\infty}|t|^{2z-1}e^{-ct^{2}}\diff{t}</syntaxhighlight> || <math>c > 0, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>(c)^(- z)* GAMMA(z) = int((abs(t))^(2*z - 1)* exp(- c*(t)^(2)), t = - infinity..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(c)^(- z)* Gamma[z] == Integrate[(Abs[t])^(2*z - 1)* Exp[- c*(t)^(2)], {t, - Infinity, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Missing Macro Error || - || -
|-
| [https://dlmf.nist.gov/5.9.E4 5.9.E4] || [[Item:Q2093|<math>\EulerGamma@{z} = \int_{1}^{\infty}t^{z-1}e^{-t}\diff{t}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)k!}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z} = \int_{1}^{\infty}t^{z-1}e^{-t}\diff{t}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)k!}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z) = int((t)^(z - 1)* exp(- t), t = 1..infinity)+ sum(((- 1)^(k))/((z + k)*factorial(k)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z] == Integrate[(t)^(z - 1)* Exp[- t], {t, 1, Infinity}, GenerateConditions->None]+ Sum[Divide[(- 1)^(k),(z + k)*(k)!], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 5]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9999999999-0.*I
Test Values: {z = 2, z = 1}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.9.E5 5.9.E5] || [[Item:Q2094|<math>\EulerGamma@{z} = \int_{0}^{\infty}t^{z-1}\left(e^{-t}-\sum_{k=0}^{n}\frac{(-1)^{k}t^{k}}{k!}\right)\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z} = \int_{0}^{\infty}t^{z-1}\left(e^{-t}-\sum_{k=0}^{n}\frac{(-1)^{k}t^{k}}{k!}\right)\diff{t}</syntaxhighlight> || <math>-n-1 < \realpart@@{z}, \realpart@@{z} < -n, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z) = int((t)^(z - 1)*(exp(- t)- sum(((- 1)^(k)* (t)^(k))/(factorial(k)), k = 0..n)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z] == Integrate[(t)^(z - 1)*(Exp[- t]- Sum[Divide[(- 1)^(k)* (t)^(k),(k)!], {k, 0, n}, GenerateConditions->None]), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Error || Skip - No test values generated
|-
| [https://dlmf.nist.gov/5.9.E6 5.9.E6] || [[Item:Q2095|<math>\EulerGamma@{z}\cos@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\cos@@{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z}\cos@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\cos@@{t}\diff{t}</syntaxhighlight> || <math>0 < \realpart@@{z}, \realpart@@{z} < 1, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z)*cos((1)/(2)*Pi*z) = int((t)^(z - 1)* cos(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z]*Cos[Divide[1,2]*Pi*z] == Integrate[(t)^(z - 1)* Cos[t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.9.E7 5.9.E7] || [[Item:Q2096|<math>\EulerGamma@{z}\sin@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\sin@@{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z}\sin@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\sin@@{t}\diff{t}</syntaxhighlight> || <math>-1 < \realpart@@{z}, \realpart@@{z} < 1, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z)*sin((1)/(2)*Pi*z) = int((t)^(z - 1)* sin(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z]*Sin[Divide[1,2]*Pi*z] == Integrate[(t)^(z - 1)* Sin[t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.9.E8 5.9.E8] || [[Item:Q2097|<math>\EulerGamma@{1+\frac{1}{n}}\cos@{\frac{\pi}{2n}} = \int_{0}^{\infty}\cos@{t^{n}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{1+\frac{1}{n}}\cos@{\frac{\pi}{2n}} = \int_{0}^{\infty}\cos@{t^{n}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(1 +(1)/(n))*cos((Pi)/(2*n)) = int(cos((t)^(n)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[1 +Divide[1,n]]*Cos[Divide[Pi,2*n]] == Integrate[Cos[(t)^(n)], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.9.E9 5.9.E9] || [[Item:Q2098|<math>\EulerGamma@{1+\frac{1}{n}}\sin@{\frac{\pi}{2n}} = \int_{0}^{\infty}\sin@{t^{n}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{1+\frac{1}{n}}\sin@{\frac{\pi}{2n}} = \int_{0}^{\infty}\sin@{t^{n}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>GAMMA(1 +(1)/(n))*sin((Pi)/(2*n)) = int(sin((t)^(n)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[1 +Divide[1,n]]*Sin[Divide[Pi,2*n]] == Integrate[Sin[(t)^(n)], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.9.E10 5.9.E10] || [[Item:Q2099|<math>\Ln@@{\EulerGamma@{z}} = \left(z-\tfrac{1}{2}\right)\ln@@{z}-z+\tfrac{1}{2}\ln@{2\pi}+2\int_{0}^{\infty}\frac{\atan@{t/z}}{e^{2\pi t}-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Ln@@{\EulerGamma@{z}} = \left(z-\tfrac{1}{2}\right)\ln@@{z}-z+\tfrac{1}{2}\ln@{2\pi}+2\int_{0}^{\infty}\frac{\atan@{t/z}}{e^{2\pi t}-1}\diff{t}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>ln(GAMMA(z)) = (z -(1)/(2))*ln(z)- z +(1)/(2)*ln(2*Pi)+ 2*int((arctan(t/z))/(exp(2*Pi*t)- 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Gamma[z]] == (z -Divide[1,2])*Log[z]- z +Divide[1,2]*Log[2*Pi]+ 2*Integrate[Divide[ArcTan[t/z],Exp[2*Pi*t]- 1], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 5] || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/5.9.E11 5.9.E11] || [[Item:Q2100|<math>\Ln@@{\EulerGamma@{z+1}} = -\EulerConstant z-\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s}}{s\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Ln@@{\EulerGamma@{z+1}} = -\EulerConstant z-\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s}}{s\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}</syntaxhighlight> || <math>\realpart@@{(z+1)} > 0</math> || <syntaxhighlight lang=mathematica>ln(GAMMA(z + 1)) = - gamma*z -(1)/(2*Pi*I)*int((Pi*(z)^(- s))/(s*sin(Pi*s))*Zeta(- s), s = - c - infinity*I..- c + infinity*I)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[Gamma[z + 1]] == - EulerGamma*z -Divide[1,2*Pi*I]*Integrate[Divide[Pi*(z)^(- s),s*Sin[Pi*s]]*Zeta[- s], {s, - c - Infinity*I, - c + Infinity*I}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .3627983593+.4645558136*I
Test Values: {c = -1.5, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.7321808519-.4375773776*I
Test Values: {c = -1.5, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.1549651868-.6096201737*I
Test Values: {c = -1.5, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.670593886e-1+1.175772123*I
Test Values: {c = -1.5, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/5.9.E12 5.9.E12] || [[Item:Q2101|<math>\digamma@{z} = \int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^{-t}}\right)\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z} = \int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^{-t}}\right)\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z) = int((exp(- t))/(t)-(exp(- z*t))/(1 - exp(- t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z] == Integrate[Divide[Exp[- t],t]-Divide[Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.9.E13 5.9.E13] || [[Item:Q2102|<math>\digamma@{z} = \ln@@{z}+\int_{0}^{\infty}\left(\frac{1}{t}-\frac{1}{1-e^{-t}}\right)e^{-tz}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z} = \ln@@{z}+\int_{0}^{\infty}\left(\frac{1}{t}-\frac{1}{1-e^{-t}}\right)e^{-tz}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z) = ln(z)+ int(((1)/(t)-(1)/(1 - exp(- t)))*exp(- t*z), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z] == Log[z]+ Integrate[(Divide[1,t]-Divide[1,1 - Exp[- t]])*Exp[- t*z], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[z, 1]}</syntaxhighlight><br></div></div>
|-
| [https://dlmf.nist.gov/5.9.E14 5.9.E14] || [[Item:Q2103|<math>\digamma@{z} = \int_{0}^{\infty}\left(e^{-t}-\frac{1}{(1+t)^{z}}\right)\frac{\diff{t}}{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z} = \int_{0}^{\infty}\left(e^{-t}-\frac{1}{(1+t)^{z}}\right)\frac{\diff{t}}{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z) = int((exp(- t)-(1)/((1 + t)^(z)))*(1)/(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z] == Integrate[(Exp[- t]-Divide[1,(1 + t)^(z)])*Divide[1,t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Skipped - Because timed out || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/5.9.E15 5.9.E15] || [[Item:Q2104|<math>\digamma@{z} = \ln@@{z}-\frac{1}{2z}-2\int_{0}^{\infty}\frac{t\diff{t}}{(t^{2}+z^{2})(e^{2\pi t}-1)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z} = \ln@@{z}-\frac{1}{2z}-2\int_{0}^{\infty}\frac{t\diff{t}}{(t^{2}+z^{2})(e^{2\pi t}-1)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z) = ln(z)-(1)/(2*z)- 2*int((t)/(((t)^(2)+ (z)^(2))*(exp(2*Pi*t)- 1)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z] == Log[z]-Divide[1,2*z]- 2*Integrate[Divide[t,((t)^(2)+ (z)^(2))*(Exp[2*Pi*t]- 1)], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .4e-10-.2711020420e-1*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.2144560970-.1791125126*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.9.E16 5.9.E16] || [[Item:Q2105|<math>\digamma@{z}+\EulerConstant = \int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z}+\EulerConstant = \int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z)+ gamma = int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z]+ EulerGamma == Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.9.E16 5.9.E16] || [[Item:Q2105|<math>\int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t} = \int_{0}^{1}\frac{1-t^{z-1}}{1-t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t} = \int_{0}^{1}\frac{1-t^{z-1}}{1-t}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity) = int((1 - (t)^(z - 1))/(1 - t), t = 0..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Divide[1 - (t)^(z - 1),1 - t], {t, 0, 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/5.9.E17 5.9.E17] || [[Item:Q2106|<math>\digamma@{z+1} = -\EulerConstant+\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s-1}}{\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{z+1} = -\EulerConstant+\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s-1}}{\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(z + 1) = - gamma +(1)/(2*Pi*I)*int((Pi*(z)^(- s - 1))/(sin(Pi*s))*Zeta(- s), s = - c - infinity*I..- c + infinity*I)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[z + 1] == - EulerGamma +Divide[1,2*Pi*I]*Integrate[Divide[Pi*(z)^(- s - 1),Sin[Pi*s]]*Zeta[- s], {s, - c - Infinity*I, - c + Infinity*I}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9666504222+.3394950970*I
Test Values: {c = -1.5, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .3622891065+1.557241225*I
Test Values: {c = -1.5, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8622891063-.6912158211*I
Test Values: {c = -1.5, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.1138310784-2.481210069*I
Test Values: {c = -1.5, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/5.9.E18 5.9.E18] || [[Item:Q2107|<math>\EulerConstant = -\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerConstant = -\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>gamma = - int(exp(- t)*ln(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerGamma == - Integrate[Exp[- t]*Log[t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.9.E18 5.9.E18] || [[Item:Q2107|<math>-\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t} = \int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t} = \int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>- int(exp(- t)*ln(t), t = 0..infinity) = int(((1)/(1 + t)- exp(- t))*(1)/(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- Integrate[Exp[- t]*Log[t], {t, 0, Infinity}, GenerateConditions->None] == Integrate[(Divide[1,1 + t]- Exp[- t])*Divide[1,t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.9.E18 5.9.E18] || [[Item:Q2107|<math>\int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t} = \int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t} = \int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(((1)/(1 + t)- exp(- t))*(1)/(t), t = 0..infinity) = int((1 - exp(- t))*(1)/(t), t = 0..1)- int(exp(- t)*(1)/(t), t = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(Divide[1,1 + t]- Exp[- t])*Divide[1,t], {t, 0, Infinity}, GenerateConditions->None] == Integrate[(1 - Exp[- t])*Divide[1,t], {t, 0, 1}, GenerateConditions->None]- Integrate[Exp[- t]*Divide[1,t], {t, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.9.E18 5.9.E18] || [[Item:Q2107|<math>\int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t} = \int_{0}^{\infty}\left(\frac{e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t} = \int_{0}^{\infty}\left(\frac{e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int((1 - exp(- t))*(1)/(t), t = 0..1)- int(exp(- t)*(1)/(t), t = 1..infinity) = int((exp(- t))/(1 - exp(- t))-(exp(- t))/(t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(1 - Exp[- t])*Divide[1,t], {t, 0, 1}, GenerateConditions->None]- Integrate[Exp[- t]*Divide[1,t], {t, 1, Infinity}, GenerateConditions->None] == Integrate[Divide[Exp[- t],1 - Exp[- t]]-Divide[Exp[- t],t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.9.E19 5.9.E19] || [[Item:Q2108|<math>\EulerGamma^{(n)}@{z} = \int_{0}^{\infty}(\ln@@{t})^{n}e^{-t}t^{z-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma^{(n)}@{z} = \int_{0}^{\infty}(\ln@@{t})^{n}e^{-t}t^{z-1}\diff{t}</syntaxhighlight> || <math>n \geq 0, \realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>diff( GAMMA(z), z$(n) ) = int((ln(t))^(n)* exp(- t)*(t)^(z - 1), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Gamma[z], {z, n}] == Integrate[(Log[t])^(n)* Exp[- t]*(t)^(z - 1), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.10#Ex1 5.10#Ex1] || [[Item:Q2110|<math>a_{0} = \tfrac{1}{12}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{0} = \tfrac{1}{12}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[0] = (1)/(12)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, 0] == Divide[1,12]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.10#Ex2 5.10#Ex2] || [[Item:Q2111|<math>a_{1} = \tfrac{1}{30}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{1} = \tfrac{1}{30}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[1] = (1)/(30)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, 1] == Divide[1,30]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.10#Ex3 5.10#Ex3] || [[Item:Q2112|<math>a_{2} = \tfrac{53}{210}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{2} = \tfrac{53}{210}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[2] = (53)/(210)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, 2] == Divide[53,210]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.10#Ex4 5.10#Ex4] || [[Item:Q2113|<math>a_{3} = \tfrac{195}{371}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{3} = \tfrac{195}{371}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[3] = (195)/(371)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, 3] == Divide[195,371]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.10#Ex5 5.10#Ex5] || [[Item:Q2114|<math>a_{4} = \tfrac{22999}{22737}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{4} = \tfrac{22999}{22737}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[4] = (22999)/(22737)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, 4] == Divide[22999,22737]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.10#Ex6 5.10#Ex6] || [[Item:Q2115|<math>a_{5} = \tfrac{299\;44523}{197\;33142}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{5} = \tfrac{299\;44523}{197\;33142}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[5] = (29944523)/(19733142)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, 5] == Divide[29944523,19733142]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.10#Ex7 5.10#Ex7] || [[Item:Q2116|<math>a_{6} = \tfrac{10\;95352\;41009}{4\;82642\;75462}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{6} = \tfrac{10\;95352\;41009}{4\;82642\;75462}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[6] = (109535241009)/(48264275462)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, 6] == Divide[109535241009,48264275462]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex1 5.11#Ex1] || [[Item:Q2120|<math>g_{0} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{0} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[0] = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 0] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex2 5.11#Ex2] || [[Item:Q2121|<math>g_{1} = \tfrac{1}{12}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{1} = \tfrac{1}{12}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[1] = (1)/(12)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 1] == Divide[1,12]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex3 5.11#Ex3] || [[Item:Q2122|<math>g_{2} = \tfrac{1}{288}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{2} = \tfrac{1}{288}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[2] = (1)/(288)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 2] == Divide[1,288]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex4 5.11#Ex4] || [[Item:Q2123|<math>g_{3} = -\tfrac{139}{51840}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{3} = -\tfrac{139}{51840}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[3] = -(139)/(51840)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 3] == -Divide[139,51840]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex5 5.11#Ex5] || [[Item:Q2124|<math>g_{4} = -\tfrac{571}{24\;88320}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{4} = -\tfrac{571}{24\;88320}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[4] = -(571)/(2488320)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 4] == -Divide[571,2488320]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex6 5.11#Ex6] || [[Item:Q2125|<math>g_{5} = \tfrac{1\;63879}{2090\;18880}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{5} = \tfrac{1\;63879}{2090\;18880}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[5] = (163879)/(209018880)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 5] == Divide[163879,209018880]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex7 5.11#Ex7] || [[Item:Q2126|<math>g_{6} = \tfrac{52\;46819}{7\;52467\;96800}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{6} = \tfrac{52\;46819}{7\;52467\;96800}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[6] = (5246819)/(75246796800)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, 6] == Divide[5246819,75246796800]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/5.11.E5 5.11.E5] || [[Item:Q2127|<math>g_{k} = \sqrt{2}\Pochhammersym{\tfrac{1}{2}}{k}a_{2k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>g_{k} = \sqrt{2}\Pochhammersym{\tfrac{1}{2}}{k}a_{2k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>g[k] = sqrt(2)*pochhammer((1)/(2), k)*a[2*k]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[g, k] == Sqrt[2]*Pochhammer[Divide[1,2], k]*Subscript[a, 2*k]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .2536529683+.1464466095*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5253324962e-1-.303300858e-1*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.430371229-.8258252140*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, k = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.112372436+.5124720135*I
Test Values: {a[2*k] = 1/2*3^(1/2)+1/2*I, g[k] = -1/2+1/2*I*3^(1/2), k = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.25365296808864424, 0.14644660940672627]
Test Values: {Rule[k, 1], Rule[Subscript[a, Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.05253324975925311, -0.03033008588991054]
Test Values: {Rule[k, 2], Rule[Subscript[a, Times[2, k]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.11.E10 5.11.E10] || [[Item:Q2132|<math>\EulerGamma@{z} = e^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{z} = e^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right)</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(z) = exp(- z)*(z)^(z)*((2*Pi)/(z))^(1/2)*(sum((g[k])/((z)^(k)), k = 0..K - 1)+ R[K](z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[z] == Exp[- z]*(z)^(z)*(Divide[2*Pi,z])^(1/2)*(Sum[Divide[Subscript[g, k],(z)^(k)], {k, 0, K - 1}, GenerateConditions->None]+ Subscript[R, K][z])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.892613380-.1706947928*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2*3^(1/2)+1/2*I, K = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.4529896033-2.955992714*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = -1/2+1/2*I*3^(1/2), K = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8926845412+1.268928985*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = 1/2-1/2*I*3^(1/2), K = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.332308320-1.516368937*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I, R[K] = 1/2*3^(1/2)+1/2*I, g[k] = -1/2*3^(1/2)-1/2*I, K = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.8926133813331316, -0.17069479199840365]
Test Values: {Rule[K, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[R, K], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.7462398809799414, -0.22409723911500246]
Test Values: {Rule[K, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[R, K], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex8 5.11#Ex8] || [[Item:Q2137|<math>G_{0}(a,b) = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>G_{0}(a,b) = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">G[0](a , b) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[G, 0][a , b] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex9 5.11#Ex9] || [[Item:Q2138|<math>G_{1}(a,b) = \tfrac{1}{2}(a-b)(a+b-1)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>G_{1}(a,b) = \tfrac{1}{2}(a-b)(a+b-1)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">G[1](a , b) = (1)/(2)*(a - b)*(a + b - 1)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[G, 1][a , b] == Divide[1,2]*(a - b)*(a + b - 1)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/5.11#Ex10 5.11#Ex10] || [[Item:Q2139|<math>G_{2}(a,b) = \frac{1}{12}\binom{a-b}{2}(3(a+b-1)^{2}-(a-b+1))</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>G_{2}(a,b) = \frac{1}{12}\binom{a-b}{2}(3(a+b-1)^{2}-(a-b+1))</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>G[2](a , b) = (1)/(12)*binomial(a - b,2)*(3*(a + b - 1)^(2)-(a - b + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[G, 2][a , b] == Divide[1,12]*Binomial[a - b,2]*(3*(a + b - 1)^(2)-(a - b + 1))</syntaxhighlight> || Failure || Failure || Error || Error
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.11#Ex11 5.11#Ex11] || [[Item:Q2140|<math>H_{0}(a,b) = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>H_{0}(a,b) = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">H[0](a , b) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[H, 0][a , b] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/5.11#Ex12 5.11#Ex12] || [[Item:Q2141|<math>H_{1}(a,b) = -\frac{1}{12}\binom{a-b}{2}(a-b+1)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>H_{1}(a,b) = -\frac{1}{12}\binom{a-b}{2}(a-b+1)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>H[1](a , b) = -(1)/(12)*binomial(a - b,2)*(a - b + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[H, 1][a , b] == -Divide[1,12]*Binomial[a - b,2]*(a - b + 1)</syntaxhighlight> || Failure || Failure || Error || Error
|-
| [https://dlmf.nist.gov/5.11#Ex13 5.11#Ex13] || [[Item:Q2142|<math>H_{2}(a,b) = \frac{1}{240}\binom{a-b}{4}(2(a-b+1)+5(a-b+1)^{2})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>H_{2}(a,b) = \frac{1}{240}\binom{a-b}{4}(2(a-b+1)+5(a-b+1)^{2})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>H[2](a , b) = (1)/(240)*binomial(a - b,4)*(2*(a - b + 1)+ 5*(a - b + 1)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[H, 2][a , b] == Divide[1,240]*Binomial[a - b,4]*(2*(a - b + 1)+ 5*(a - b + 1)^(2))</syntaxhighlight> || Failure || Failure || Error || Error
|-
| [https://dlmf.nist.gov/5.12.E1 5.12.E1] || [[Item:Q2146|<math>\EulerBeta@{a}{b} = \int_{0}^{1}t^{a-1}(1-t)^{b-1}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerBeta@{a}{b} = \int_{0}^{1}t^{a-1}(1-t)^{b-1}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Beta(a, b) = int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Beta[a, b] == Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [11 / 36]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -2]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, 1.5], Rule[b, -2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.12.E1 5.12.E1] || [[Item:Q2146|<math>\int_{0}^{1}t^{a-1}(1-t)^{b-1}\diff{t} = \frac{\EulerGamma@{a}\EulerGamma@{b}}{\EulerGamma@{a+b}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}t^{a-1}(1-t)^{b-1}\diff{t} = \frac{\EulerGamma@{a}\EulerGamma@{b}}{\EulerGamma@{a+b}}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{(a+b)} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..1) = (GAMMA(a)*GAMMA(b))/(GAMMA(a + b))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, 1}, GenerateConditions->None] == Divide[Gamma[a]*Gamma[b],Gamma[a + b]]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 9]
|-
| [https://dlmf.nist.gov/5.12.E2 5.12.E2] || [[Item:Q2147|<math>\int_{0}^{\pi/2}\sin^{2a-1}@@{\theta}\cos^{2b-1}@@{\theta}\diff{\theta} = \tfrac{1}{2}\EulerBeta@{a}{b}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\pi/2}\sin^{2a-1}@@{\theta}\cos^{2b-1}@@{\theta}\diff{\theta} = \tfrac{1}{2}\EulerBeta@{a}{b}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{(a+b)} > 0</math> || <syntaxhighlight lang=mathematica>int((sin(theta))^(2*a - 1)* (cos(theta))^(2*b - 1), theta = 0..Pi/2) = (1)/(2)*Beta(a, b)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(Sin[\[Theta]])^(2*a - 1)* (Cos[\[Theta]])^(2*b - 1), {\[Theta], 0, Pi/2}, GenerateConditions->None] == Divide[1,2]*Beta[a, b]</syntaxhighlight> || Failure || Successful || Error || Successful [Tested: 9]
|-
| [https://dlmf.nist.gov/5.12.E3 5.12.E3] || [[Item:Q2148|<math>\int_{0}^{\infty}\frac{t^{a-1}\diff{t}}{(1+t)^{a+b}} = \EulerBeta@{a}{b}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\frac{t^{a-1}\diff{t}}{(1+t)^{a+b}} = \EulerBeta@{a}{b}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{(a+b)} > 0</math> || <syntaxhighlight lang=mathematica>int(((t)^(a - 1))/((1 + t)^(a + b)), t = 0..infinity) = Beta(a, b)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[(t)^(a - 1),(1 + t)^(a + b)], {t, 0, Infinity}, GenerateConditions->None] == Beta[a, b]</syntaxhighlight> || Failure || Successful || Error || Successful [Tested: 9]
|-
| [https://dlmf.nist.gov/5.12.E4 5.12.E4] || [[Item:Q2149|<math>\int_{0}^{1}\frac{t^{a-1}(1-t)^{b-1}}{(t+z)^{a+b}}\diff{t} = \EulerBeta@{a}{b}(1+z)^{-a}z^{-b}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{1}\frac{t^{a-1}(1-t)^{b-1}}{(t+z)^{a+b}}\diff{t} = \EulerBeta@{a}{b}(1+z)^{-a}z^{-b}</syntaxhighlight> || <math>|\phase@@{z}| < \pi, \realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{(a+b)} > 0</math> || <syntaxhighlight lang=mathematica>int(((t)^(a - 1)*(1 - t)^(b - 1))/((t + z)^(a + b)), t = 0..1) = Beta(a, b)*(1 + z)^(- a)* (z)^(- b)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[(t)^(a - 1)*(1 - t)^(b - 1),(t + z)^(a + b)], {t, 0, 1}, GenerateConditions->None] == Beta[a, b]*(1 + z)^(- a)* (z)^(- b)</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [77 / 252]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.12.E5 5.12.E5] || [[Item:Q2150|<math>\int_{0}^{\pi/2}(\cos@@{t})^{a-1}\cos@{bt}\diff{t} = \frac{\pi}{2^{a}}\frac{1}{a\EulerBeta@{\frac{1}{2}(a+b+1)}{\frac{1}{2}(a-b+1)}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\pi/2}(\cos@@{t})^{a-1}\cos@{bt}\diff{t} = \frac{\pi}{2^{a}}\frac{1}{a\EulerBeta@{\frac{1}{2}(a+b+1)}{\frac{1}{2}(a-b+1)}}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{(\frac{1}{2}(a+b+1))} > 0, \realpart@@{(\frac{1}{2}(a-b+1))} > 0, \realpart@@{((\frac{1}{2}(a+b+1))+b)} > 0, \realpart@@{(a+(\frac{1}{2}(a-b+1)))} > 0</math> || <syntaxhighlight lang=mathematica>int((cos(t))^(a - 1)* cos(b*t), t = 0..Pi/2) = (Pi)/((2)^(a))*(1)/(a*Beta((1)/(2)*(a + b + 1), (1)/(2)*(a - b + 1)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(Cos[t])^(a - 1)* Cos[b*t], {t, 0, Pi/2}, GenerateConditions->None] == Divide[Pi,(2)^(a)]*Divide[1,a*Beta[Divide[1,2]*(a + b + 1), Divide[1,2]*(a - b + 1)]]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 18] || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/5.12.E6 5.12.E6] || [[Item:Q2151|<math>\int_{0}^{\pi}(\sin@@{t})^{a-1}e^{ibt}\diff{t} = \frac{\pi}{2^{a-1}}\frac{e^{i\pi b/2}}{a\EulerBeta@{\frac{1}{2}(a+b+1)}{\frac{1}{2}(a-b+1)}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\pi}(\sin@@{t})^{a-1}e^{ibt}\diff{t} = \frac{\pi}{2^{a-1}}\frac{e^{i\pi b/2}}{a\EulerBeta@{\frac{1}{2}(a+b+1)}{\frac{1}{2}(a-b+1)}}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{(\frac{1}{2}(a+b+1))} > 0, \realpart@@{(\frac{1}{2}(a-b+1))} > 0, \realpart@@{((\frac{1}{2}(a+b+1))+b)} > 0, \realpart@@{(a+(\frac{1}{2}(a-b+1)))} > 0</math> || <syntaxhighlight lang=mathematica>int((sin(t))^(a - 1)* exp(I*b*t), t = 0..Pi) = (Pi)/((2)^(a - 1))*(exp(I*Pi*b/2))/(a*Beta((1)/(2)*(a + b + 1), (1)/(2)*(a - b + 1)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(Sin[t])^(a - 1)* Exp[I*b*t], {t, 0, Pi}, GenerateConditions->None] == Divide[Pi,(2)^(a - 1)]*Divide[Exp[I*Pi*b/2],a*Beta[Divide[1,2]*(a + b + 1), Divide[1,2]*(a - b + 1)]]</syntaxhighlight> || Failure || Aborted || Successful [Tested: 18] || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
Test Values: {Rule[a, 1.5], Rule[b, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: DirectedInfinity[]
Test Values: {Rule[a, 1.5], Rule[b, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.12.E7 5.12.E7] || [[Item:Q2152|<math>\int_{0}^{\infty}\frac{\cosh@{2bt}}{(\cosh@@{t})^{2a}}\diff{t} = 4^{a-1}\EulerBeta@{a+b}{a-b}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}\frac{\cosh@{2bt}}{(\cosh@@{t})^{2a}}\diff{t} = 4^{a-1}\EulerBeta@{a+b}{a-b}</syntaxhighlight> || <math>\realpart@@{a} > |\realpart@@{b}|, \realpart@@{(a+b)} > 0, \realpart@@{(a-b)} > 0, \realpart@@{((a+b)+b)} > 0, \realpart@@{(a+(a-b))} > 0</math> || <syntaxhighlight lang=mathematica>int((cosh(2*b*t))/((cosh(t))^(2*a)), t = 0..infinity) = (4)^(a - 1)* Beta(a + b, a - b)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[Cosh[2*b*t],(Cosh[t])^(2*a)], {t, 0, Infinity}, GenerateConditions->None] == (4)^(a - 1)* Beta[a + b, a - b]</syntaxhighlight> || Failure || Failure || Successful [Tested: 6] || Successful [Tested: 6]
|-
| [https://dlmf.nist.gov/5.12.E8 5.12.E8] || [[Item:Q2153|<math>\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\diff{t}}{(w+it)^{a}(z-it)^{b}} = \frac{(w+z)^{1-a-b}}{(a+b-1)\EulerBeta@{a}{b}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\diff{t}}{(w+it)^{a}(z-it)^{b}} = \frac{(w+z)^{1-a-b}}{(a+b-1)\EulerBeta@{a}{b}}</syntaxhighlight> || <math>\realpart@{a+b} > 1, \realpart@@{w} > 0, \realpart@@{z} > 0, \realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{(a+b)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(2*Pi)*int((1)/((w + I*t)^(a)*(z - I*t)^(b)), t = - infinity..infinity) = ((w + z)^(1 - a - b))/((a + b - 1)*Beta(a, b))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2*Pi]*Integrate[Divide[1,(w + I*t)^(a)*(z - I*t)^(b)], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[(w + z)^(1 - a - b),(a + b - 1)*Beta[a, b]]</syntaxhighlight> || Skipped - Unable to analyze test case: Null || Failure || - || Successful [Tested: 250]
|-
| [https://dlmf.nist.gov/5.12.E9 5.12.E9] || [[Item:Q2154|<math>\frac{1}{2\pi i}\int_{c-\infty i}^{c+\infty i}t^{-a}(1-t)^{-1-b}\diff{t} = \frac{1}{b\EulerBeta@{a}{b}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{2\pi i}\int_{c-\infty i}^{c+\infty i}t^{-a}(1-t)^{-1-b}\diff{t} = \frac{1}{b\EulerBeta@{a}{b}}</syntaxhighlight> || <math>0 < c, c < 1, \realpart@{a+b} > 0, \realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{(a+b)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(2*Pi*I)*int((t)^(- a)*(1 - t)^(- 1 - b), t = c - infinity*I..c + infinity*I) = (1)/(b*Beta(a, b))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2*Pi*I]*Integrate[(t)^(- a)*(1 - t)^(- 1 - b), {t, c - Infinity*I, c + Infinity*I}, GenerateConditions->None] == Divide[1,b*Beta[a, b]]</syntaxhighlight> || Skipped - Unable to analyze test case: Null || Aborted || - || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/5.12.E10 5.12.E10] || [[Item:Q2155|<math>\frac{1}{2\pi i}\int_{0}^{(1+)}t^{a-1}(t-1)^{b-1}\diff{t} = \frac{\sin@{\pi b}}{\pi}\EulerBeta@{a}{b}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{2\pi i}\int_{0}^{(1+)}t^{a-1}(t-1)^{b-1}\diff{t} = \frac{\sin@{\pi b}}{\pi}\EulerBeta@{a}{b}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{(a+b)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(2*Pi*I)*int((t)^(a - 1)*(t - 1)^(b - 1), t = 0..(1 +)) = (sin(Pi*b))/(Pi)*Beta(a, b)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2*Pi*I]*Integrate[(t)^(a - 1)*(t - 1)^(b - 1), {t, 0, (1 +)}, GenerateConditions->None] == Divide[Sin[Pi*b],Pi]*Beta[a, b]</syntaxhighlight> || Skipped - Unable to analyze test case: Null || Failure || - || Error
|-
| [https://dlmf.nist.gov/5.12.E11 5.12.E11] || [[Item:Q2156|<math>\frac{1}{e^{2\pi ia}-1}\int_{\infty}^{(0+)}t^{a-1}(1+t)^{-a-b}\diff{t} = \EulerBeta@{a}{b}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{e^{2\pi ia}-1}\int_{\infty}^{(0+)}t^{a-1}(1+t)^{-a-b}\diff{t} = \EulerBeta@{a}{b}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{(a+b)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(exp(2*Pi*I*a)- 1)*int((t)^(a - 1)*(1 + t)^(- a - b), t = infinity..(0 +)) = Beta(a, b)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Exp[2*Pi*I*a]- 1]*Integrate[(t)^(a - 1)*(1 + t)^(- a - b), {t, Infinity, (0 +)}, GenerateConditions->None] == Beta[a, b]</syntaxhighlight> || Error || Failure || - || Error
|-
| [https://dlmf.nist.gov/5.12.E12 5.12.E12] || [[Item:Q2157|<math>\int_{P}^{(1+,0+,1-,0-)}t^{a-1}(1-t)^{b-1}\diff{t} = -4e^{\pi i(a+b)}\sin@{\pi a}\sin@{\pi b}\EulerBeta@{a}{b}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{P}^{(1+,0+,1-,0-)}t^{a-1}(1-t)^{b-1}\diff{t} = -4e^{\pi i(a+b)}\sin@{\pi a}\sin@{\pi b}\EulerBeta@{a}{b}</syntaxhighlight> || <math>\realpart@@{a} > 0, \realpart@@{b} > 0, \realpart@@{(a+b)} > 0</math> || <syntaxhighlight lang=mathematica>int((t)^(a - 1)*(1 - t)^(b - 1), t = P..(1 + , 0 + , 1 - , 0 -)) = - 4*exp(Pi*I*(a + b))*sin(Pi*a)*sin(Pi*b)*Beta(a, b)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, P, (1 + , 0 + , 1 - , 0 -)}, GenerateConditions->None] == - 4*Exp[Pi*I*(a + b)]*Sin[Pi*a]*Sin[Pi*b]*Beta[a, b]</syntaxhighlight> || Error || Failure || - || Error
|-
| [https://dlmf.nist.gov/5.13.E1 5.13.E1] || [[Item:Q2158|<math>\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\EulerGamma@{s+a}\EulerGamma@{b-s}z^{-s}\diff{s} = \frac{\EulerGamma@{a+b}z^{a}}{(1+z)^{a+b}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\EulerGamma@{s+a}\EulerGamma@{b-s}z^{-s}\diff{s} = \frac{\EulerGamma@{a+b}z^{a}}{(1+z)^{a+b}}</syntaxhighlight> || <math>\realpart@{a+b} > 0, -\realpart@@{a} < c, c < \realpart@@{b}, |\phase@@{z}| < \pi, \realpart@@{(s+a)} > 0, \realpart@@{(b-s)} > 0, \realpart@@{(a+b)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(2*Pi*I)*int(GAMMA(s + a)*GAMMA(b - s)*(z)^(- s), s = c - I*infinity..c + I*infinity) = (GAMMA(a + b)*(z)^(a))/((1 + z)^(a + b))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2*Pi*I]*Integrate[Gamma[s + a]*Gamma[b - s]*(z)^(- s), {s, c - I*Infinity, c + I*Infinity}, GenerateConditions->None] == Divide[Gamma[a + b]*(z)^(a),(1 + z)^(a + b)]</syntaxhighlight> || Skipped - Unable to analyze test case: Null || Aborted || - || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/5.13.E2 5.13.E2] || [[Item:Q2159|<math>\frac{1}{2\pi}\int_{-\infty}^{\infty}|\EulerGamma@{a+it}|^{2}e^{(2b-\pi)t}\diff{t} = \frac{\EulerGamma@{2a}}{(2\sin@@{b})^{2a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{2\pi}\int_{-\infty}^{\infty}|\EulerGamma@{a+it}|^{2}e^{(2b-\pi)t}\diff{t} = \frac{\EulerGamma@{2a}}{(2\sin@@{b})^{2a}}</syntaxhighlight> || <math>a > 0, 0 < b, b < \pi, \realpart@@{(a+\iunit t)} > 0, \realpart@@{(2a)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(2*Pi)*int((abs(GAMMA(a + I*t)))^(2)* exp((2*b - Pi)*t), t = - infinity..infinity) = (GAMMA(2*a))/((2*sin(b))^(2*a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2*Pi]*Integrate[(Abs[Gamma[a + I*t]])^(2)* Exp[(2*b - Pi)*t], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[Gamma[2*a],(2*Sin[b])^(2*a)]</syntaxhighlight> || Skipped - Unable to analyze test case: Null || Aborted || - || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/5.13.E3 5.13.E3] || [[Item:Q2160|<math>\frac{1}{2\pi}\int_{-\infty}^{\infty}\EulerGamma@{a+it}\EulerGamma@{b+it}\EulerGamma@{c-it}\EulerGamma@{d-it}\diff{t} = \frac{\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b+c}\EulerGamma@{b+d}}{\EulerGamma@{a+b+c+d}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{2\pi}\int_{-\infty}^{\infty}\EulerGamma@{a+it}\EulerGamma@{b+it}\EulerGamma@{c-it}\EulerGamma@{d-it}\diff{t} = \frac{\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b+c}\EulerGamma@{b+d}}{\EulerGamma@{a+b+c+d}}</syntaxhighlight> || <math>\realpart@@{(a+\iunit t)} > 0, \realpart@@{(b+\iunit t)} > 0, \realpart@@{(c-\iunit t)} > 0, \realpart@@{(d-\iunit t)} > 0, \realpart@@{(a+c)} > 0, \realpart@@{(a+d)} > 0, \realpart@@{(b+c)} > 0, \realpart@@{(b+d)} > 0, \realpart@@{(a+b+c+d)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(2*Pi)*int(GAMMA(a + I*t)*GAMMA(b + I*t)*GAMMA(c - I*t)*GAMMA(d - I*t), t = - infinity..infinity) = (GAMMA(a + c)*GAMMA(a + d)*GAMMA(b + c)*GAMMA(b + d))/(GAMMA(a + b + c + d))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2*Pi]*Integrate[Gamma[a + I*t]*Gamma[b + I*t]*Gamma[c - I*t]*Gamma[d - I*t], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[Gamma[a + c]*Gamma[a + d]*Gamma[b + c]*Gamma[b + d],Gamma[a + b + c + d]]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/5.13.E4 5.13.E4] || [[Item:Q2161|<math>\int_{-\infty}^{\infty}\frac{\diff{t}}{\EulerGamma@{a+t}\EulerGamma@{b+t}\EulerGamma@{c-t}\EulerGamma@{d-t}} = \frac{\EulerGamma@{a+b+c+d-3}}{\EulerGamma@{a+c-1}\EulerGamma@{a+d-1}\EulerGamma@{b+c-1}\EulerGamma@{b+d-1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-\infty}^{\infty}\frac{\diff{t}}{\EulerGamma@{a+t}\EulerGamma@{b+t}\EulerGamma@{c-t}\EulerGamma@{d-t}} = \frac{\EulerGamma@{a+b+c+d-3}}{\EulerGamma@{a+c-1}\EulerGamma@{a+d-1}\EulerGamma@{b+c-1}\EulerGamma@{b+d-1}}</syntaxhighlight> || <math>\realpart@{a+b+c+d} > 3, \realpart@@{(a+t)} > 0, \realpart@@{(b+t)} > 0, \realpart@@{(c-t)} > 0, \realpart@@{(d-t)} > 0, \realpart@@{(a+b+c+d-3)} > 0, \realpart@@{(a+c-1)} > 0, \realpart@@{(a+d-1)} > 0, \realpart@@{(b+c-1)} > 0, \realpart@@{(b+d-1)} > 0</math> || <syntaxhighlight lang=mathematica>int((1)/(GAMMA(a + t)*GAMMA(b + t)*GAMMA(c - t)*GAMMA(d - t)), t = - infinity..infinity) = (GAMMA(a + b + c + d - 3))/(GAMMA(a + c - 1)*GAMMA(a + d - 1)*GAMMA(b + c - 1)*GAMMA(b + d - 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Divide[1,Gamma[a + t]*Gamma[b + t]*Gamma[c - t]*Gamma[d - t]], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[Gamma[a + b + c + d - 3],Gamma[a + c - 1]*Gamma[a + d - 1]*Gamma[b + c - 1]*Gamma[b + d - 1]]</syntaxhighlight> || Failure || Aborted || Manual Skip! || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/5.13.E5 5.13.E5] || [[Item:Q2162|<math>\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\prod_{k=1}^{4}\EulerGamma@{a_{k}+it}\EulerGamma@{a_{k}-it}}{\EulerGamma@{2it}\EulerGamma@{-2it}}\diff{t} = \frac{\prod_{1\leq j<k\leq 4}\EulerGamma@{a_{j}+a_{k}}}{\EulerGamma@{a_{1}+a_{2}+a_{3}+a_{4}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\prod_{k=1}^{4}\EulerGamma@{a_{k}+it}\EulerGamma@{a_{k}-it}}{\EulerGamma@{2it}\EulerGamma@{-2it}}\diff{t} = \frac{\prod_{1\leq j<k\leq 4}\EulerGamma@{a_{j}+a_{k}}}{\EulerGamma@{a_{1}+a_{2}+a_{3}+a_{4}}}</syntaxhighlight> || <math>\realpart@{a_{k}} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(4*Pi)*int((product(GAMMA(a[k]+ I*t)*GAMMA(a[k]- I*t), k = 1..4))/(GAMMA(2*I*t)*GAMMA(- 2*I*t)), t = - infinity..infinity) = (product(product(GAMMA(a[j]+ a[k]), k = j + 1..4), j = 1..k - 1))/(GAMMA(a[1]+ a[2]+ a[3]+ a[4]))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,4*Pi]*Integrate[Divide[Product[Gamma[Subscript[a, k]+ I*t]*Gamma[Subscript[a, k]- I*t], {k, 1, 4}, GenerateConditions->None],Gamma[2*I*t]*Gamma[- 2*I*t]], {t, - Infinity, Infinity}, GenerateConditions->None] == Divide[Product[Product[Gamma[Subscript[a, j]+ Subscript[a, k]], {k, j + 1, 4}, GenerateConditions->None], {j, 1, k - 1}, GenerateConditions->None],Gamma[Subscript[a, 1]+ Subscript[a, 2]+ Subscript[a, 3]+ Subscript[a, 4]]]</syntaxhighlight> || Error || Aborted || - || Skipped - Because timed out
|-
| [https://dlmf.nist.gov/5.15.E1 5.15.E1] || [[Item:Q2170|<math>\digamma'@{z} = \sum_{k=0}^{\infty}\frac{1}{(k+z)^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma'@{z} = \sum_{k=0}^{\infty}\frac{1}{(k+z)^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff( Psi(z), z$(1) ) = sum((1)/((k + z)^(2)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[PolyGamma[z], {z, 1}] == Sum[Divide[1,(k + z)^(2)], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 0] || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.15.E2 5.15.E2] || [[Item:Q2171|<math>\polygamma{n}@{1} = (-1)^{n+1}n!\Riemannzeta@{n+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\polygamma{n}@{1} = (-1)^{n+1}n!\Riemannzeta@{n+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(n, 1) = (- 1)^(n + 1)* factorial(n)*Zeta(n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[n, 1] == (- 1)^(n + 1)* (n)!*Zeta[n + 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.15.E3 5.15.E3] || [[Item:Q2172|<math>\polygamma{n}@{\tfrac{1}{2}} = (-1)^{n+1}n!(2^{n+1}-1)\Riemannzeta@{n+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\polygamma{n}@{\tfrac{1}{2}} = (-1)^{n+1}n!(2^{n+1}-1)\Riemannzeta@{n+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi(n, (1)/(2)) = (- 1)^(n + 1)* factorial(n)*((2)^(n + 1)- 1)*Zeta(n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[n, Divide[1,2]] == (- 1)^(n + 1)* (n)!*((2)^(n + 1)- 1)*Zeta[n + 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.15.E4 5.15.E4] || [[Item:Q2173|<math>\digamma'@{n-\tfrac{1}{2}} = \tfrac{1}{2}\pi^{2}-4\sum_{k=1}^{n-1}\frac{1}{(2k-1)^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma'@{n-\tfrac{1}{2}} = \tfrac{1}{2}\pi^{2}-4\sum_{k=1}^{n-1}\frac{1}{(2k-1)^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>subs( temp=n -(1)/(2), diff( Psi(temp), temp$(1) ) ) = (1)/(2)*(Pi)^(2)- 4*sum((1)/((2*k - 1)^(2)), k = 1..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[PolyGamma[temp], {temp, 1}]/.temp-> n -Divide[1,2]) == Divide[1,2]*(Pi)^(2)- 4*Sum[Divide[1,(2*k - 1)^(2)], {k, 1, n - 1}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
|-
| [https://dlmf.nist.gov/5.15.E5 5.15.E5] || [[Item:Q2174|<math>\digamma^{(n)}@{z+1} = \digamma^{(n)}@{z}+(-1)^{n}n!z^{-n-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma^{(n)}@{z+1} = \digamma^{(n)}@{z}+(-1)^{n}n!z^{-n-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>subs( temp=z + 1, diff( Psi(temp), temp$(n) ) ) = diff( Psi(z), z$(n) )+(- 1)^(n)* factorial(n)*(z)^(- n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[PolyGamma[temp], {temp, n}]/.temp-> z + 1) == D[PolyGamma[z], {z, n}]+(- 1)^(n)* (n)!*(z)^(- n - 1)</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
|-
| [https://dlmf.nist.gov/5.15.E6 5.15.E6] || [[Item:Q2175|<math>\digamma^{(n)}@{1-z}+(-1)^{n-1}\digamma^{(n)}@{z} = (-1)^{n}\pi\deriv[n]{}{z}\cot@{\pi z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma^{(n)}@{1-z}+(-1)^{n-1}\digamma^{(n)}@{z} = (-1)^{n}\pi\deriv[n]{}{z}\cot@{\pi z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>subs( temp=1 - z, diff( Psi(temp), temp$(n) ) )+(- 1)^(n - 1)* diff( Psi(z), z$(n) ) = (- 1)^(n)* Pi*diff(cot(Pi*z), [z$(n)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[PolyGamma[temp], {temp, n}]/.temp-> 1 - z)+(- 1)^(n - 1)* D[PolyGamma[z], {z, n}] == (- 1)^(n)* Pi*D[Cot[Pi*z], {z, n}]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-1.111486978443634, 1.4242909397222407], Times[Complex[-1.1253971041044755, 1.3474673991212198], Inactive[Sum][Times[Power[-0.5, K[1.0]], Power[Complex[2.0570132833277626, -0.06826349589921218], K[1.0]], Factorial[K[1.0]], StirlingS2[1.0, K[1.0]]]
Test Values: {K[1.0], 0.0, 1.0}]]], {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[9.936030880873925, 6.770945349247037], Times[Complex[-8.466387364061939, -7.071078549251696], Inactive[Sum][Times[Power[-0.5, K[1.0]], Power[Complex[2.0570132833277626, -0.06826349589921218], K[1.0]], Factorial[K[1.0]], StirlingS2[2.0, K[1.0]]]
Test Values: {K[1.0], 0.0, 2.0}]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.15.E7 5.15.E7] || [[Item:Q2176|<math>\digamma^{(n)}@{mz} = \frac{1}{m^{n+1}}\sum_{k=0}^{m-1}\digamma^{(n)}@{z+\frac{k}{m}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma^{(n)}@{mz} = \frac{1}{m^{n+1}}\sum_{k=0}^{m-1}\digamma^{(n)}@{z+\frac{k}{m}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff( Psi(m*z), m*z$(n) ) = (1)/((m)^(n + 1))*sum(subs( temp=z +(k)/(m), diff( Psi(temp), temp$(n) ) ), k = 0..m - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[PolyGamma[m*z], {m*z, n}] == Divide[1,(m)^(n + 1)]*Sum[D[PolyGamma[temp], {temp, n}]/.temp-> z +Divide[k,m], {k, 0, m - 1}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [63 / 63]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-1.1320242650810568, 1.0823171404691536], D[Complex[-0.4765906465900115, 0.839495097073875]
Test Values: {Complex[0.8660254037844387, 0.49999999999999994], 1.0}]], {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.3478434500030721, -2.260508246850942], D[Complex[-0.4765906465900115, 0.839495097073875]
Test Values: {Complex[0.8660254037844387, 0.49999999999999994], 2.0}]], {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.16.E1 5.16.E1] || [[Item:Q2178|<math>\sum_{k=1}^{\infty}(-1)^{k}\digamma'@{k} = -\frac{\pi^{2}}{8}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=1}^{\infty}(-1)^{k}\digamma'@{k} = -\frac{\pi^{2}}{8}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((- 1)^(k)* diff( Psi(k), k$(1) ), k = 1..infinity) = -((Pi)^(2))/(8)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(- 1)^(k)* D[PolyGamma[k], {k, 1}], {k, 1, Infinity}, GenerateConditions->None] == -Divide[(Pi)^(2),8]</syntaxhighlight> || Failure || Successful || Successful [Tested: 0] || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.16.E2 5.16.E2] || [[Item:Q2179|<math>\sum_{k=1}^{\infty}\frac{1}{k}\digamma'@{k+1} = \Riemannzeta@{3}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=1}^{\infty}\frac{1}{k}\digamma'@{k+1} = \Riemannzeta@{3}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((1)/(k)*subs( temp=k + 1, diff( Psi(temp), temp$(1) ) ), k = 1..infinity) = Zeta(3)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[Divide[1,k]*(D[PolyGamma[temp], {temp, 1}]/.temp-> k + 1), {k, 1, Infinity}, GenerateConditions->None] == Zeta[3]</syntaxhighlight> || Failure || Successful || Successful [Tested: 0] || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.16.E2 5.16.E2] || [[Item:Q2179|<math>\Riemannzeta@{3} = -\frac{1}{2}\digamma''@{1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta@{3} = -\frac{1}{2}\digamma''@{1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Zeta(3) = -(1)/(2)*subs( temp=1, diff( Psi(temp), temp$(2) ) )</syntaxhighlight> || <syntaxhighlight lang=mathematica>Zeta[3] == -Divide[1,2]*(D[PolyGamma[temp], {temp, 2}]/.temp-> 1)</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.17#Ex1 5.17#Ex1] || [[Item:Q2180|<math>\BarnesG@{z+1} = \EulerGamma@{z}\BarnesG@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BarnesG@{z+1} = \EulerGamma@{z}\BarnesG@{z}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BarnesG[z + 1] == Gamma[z]*BarnesG[z]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 5]
|-
| [https://dlmf.nist.gov/5.17#Ex2 5.17#Ex2] || [[Item:Q2181|<math>\BarnesG@{1} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BarnesG@{1} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BarnesG[1] == 1</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.17.E3 5.17.E3] || [[Item:Q2183|<math>\BarnesG@{z+1} = (2\pi)^{z/2}\exp@{-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\EulerConstant z^{2}}\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp@{-z+\frac{z^{2}}{2k}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BarnesG@{z+1} = (2\pi)^{z/2}\exp@{-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\EulerConstant z^{2}}\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp@{-z+\frac{z^{2}}{2k}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>BarnesG[z + 1] == (2*Pi)^(z/2)* Exp[-Divide[1,2]*z*(z + 1)-Divide[1,2]*EulerGamma*(z)^(2)]* Product[(1 +Divide[z,k])^(k)* Exp[- z +Divide[(z)^(2),2*k]], {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 7]
|-
| [https://dlmf.nist.gov/5.17.E4 5.17.E4] || [[Item:Q2184|<math>\Ln@@{\BarnesG@{z+1}} = \tfrac{1}{2}z\ln@{2\pi}-\tfrac{1}{2}z(z+1)+z\Ln@@{\EulerGamma@{z+1}}-\int_{0}^{z}\Ln@@{\EulerGamma@{t+1}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Ln@@{\BarnesG@{z+1}} = \tfrac{1}{2}z\ln@{2\pi}-\tfrac{1}{2}z(z+1)+z\Ln@@{\EulerGamma@{z+1}}-\int_{0}^{z}\Ln@@{\EulerGamma@{t+1}}\diff{t}</syntaxhighlight> || <math>\realpart@@{(z+1)} > 0, \realpart@@{(t+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Log[BarnesG[z + 1]] == Divide[1,2]*z*Log[2*Pi]-Divide[1,2]*z*(z + 1)+ z*Log[Gamma[z + 1]]- Integrate[Log[Gamma[t + 1]], {t, 0, z}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 7]
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.17.E6 5.17.E6] || [[Item:Q2186|<math>A = e^{C}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>A = e^{C}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">A = exp(C)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">A == Exp[C]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/5.17.E7 5.17.E7] || [[Item:Q2187|<math>C = \lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln@@{k}-\left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln@@{n}+\tfrac{1}{4}n^{2}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>C = \lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln@@{k}-\left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln@@{n}+\tfrac{1}{4}n^{2}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>C = limit(sum(k*ln(k), k = 1..n)-((1)/(2)*(n)^(2)+(1)/(2)*n +(1)/(12))*ln(n)+(1)/(4)*(n)^(2), n = infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>C == Limit[Sum[k*Log[k], {k, 1, n}, GenerateConditions->None]-(Divide[1,2]*(n)^(2)+Divide[1,2]*n +Divide[1,12])*Log[n]+Divide[1,4]*(n)^(2), n -> Infinity, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .6172709270+.5000000000*I
Test Values: {C = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.7487544770+.8660254040*I
Test Values: {C = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2512455230-.8660254040*I
Test Values: {C = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.114779881-.5000000000*I
Test Values: {C = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.6172709267506544, 0.49999999999999994]
Test Values: {Rule[C, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.7487544770337841, 0.8660254037844387]
Test Values: {Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.17.E7 5.17.E7] || [[Item:Q2187|<math>\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln@@{k}-\left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln@@{n}+\tfrac{1}{4}n^{2}\right) = \frac{\EulerConstant+\ln@{2\pi}}{12}-\frac{\Riemannzeta'@{2}}{2\pi^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln@@{k}-\left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln@@{n}+\tfrac{1}{4}n^{2}\right) = \frac{\EulerConstant+\ln@{2\pi}}{12}-\frac{\Riemannzeta'@{2}}{2\pi^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>limit(sum(k*ln(k), k = 1..n)-((1)/(2)*(n)^(2)+(1)/(2)*n +(1)/(12))*ln(n)+(1)/(4)*(n)^(2), n = infinity) = (gamma + ln(2*Pi))/(12)-(subs( temp=2, diff( Zeta(temp), temp$(1) ) ))/(2*(Pi)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Limit[Sum[k*Log[k], {k, 1, n}, GenerateConditions->None]-(Divide[1,2]*(n)^(2)+Divide[1,2]*n +Divide[1,12])*Log[n]+Divide[1,4]*(n)^(2), n -> Infinity, GenerateConditions->None] == Divide[EulerGamma + Log[2*Pi],12]-Divide[D[Zeta[temp], {temp, 1}]/.temp-> 2,2*(Pi)^(2)]</syntaxhighlight> || Failure || Successful || Successful [Tested: 0] || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.17.E7 5.17.E7] || [[Item:Q2187|<math>\frac{\EulerConstant+\ln@{2\pi}}{12}-\frac{\Riemannzeta'@{2}}{2\pi^{2}} = \frac{1}{12}-\Riemannzeta'@{-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{\EulerConstant+\ln@{2\pi}}{12}-\frac{\Riemannzeta'@{2}}{2\pi^{2}} = \frac{1}{12}-\Riemannzeta'@{-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(gamma + ln(2*Pi))/(12)-(subs( temp=2, diff( Zeta(temp), temp$(1) ) ))/(2*(Pi)^(2)) = (1)/(12)- subs( temp=- 1, diff( Zeta(temp), temp$(1) ) )</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[EulerGamma + Log[2*Pi],12]-Divide[D[Zeta[temp], {temp, 1}]/.temp-> 2,2*(Pi)^(2)] == Divide[1,12]- (D[Zeta[temp], {temp, 1}]/.temp-> - 1)</syntaxhighlight> || Failure || Successful || Successful [Tested: 0] || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.18.E1 5.18.E1] || [[Item:Q2188|<math>\qPochhammer{a}{q}{n} = \prod_{k=0}^{n-1}(1-aq^{k})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\qPochhammer{a}{q}{n} = \prod_{k=0}^{n-1}(1-aq^{k})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>QPochhammer(a, q, n) = product(1 - a*(q)^(k), k = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>QPochhammer[a, q, n] == Product[1 - a*(q)^(k), {k, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 60]
|-
| [https://dlmf.nist.gov/5.18.E3 5.18.E3] || [[Item:Q2190|<math>\qPochhammer{a}{q}{\infty} = \prod_{k=0}^{\infty}(1-aq^{k})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\qPochhammer{a}{q}{\infty} = \prod_{k=0}^{\infty}(1-aq^{k})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>QPochhammer(a, q, infinity) = product(1 - a*(q)^(k), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>QPochhammer[a, q, Infinity] == Product[1 - a*(q)^(k), {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [48 / 60]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[-1.0, QPochhammer[-1.5, Complex[0.8660254037844387, 0.49999999999999994]]], QPochhammer[-1.5, Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]]
Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.18.E4 5.18.E4] || [[Item:Q2191|<math>\qGamma{q}@{z} = \qPochhammer{q}{q}{\infty}(1-q)^{1-z}/\qPochhammer{q^{z}}{q}{\infty}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\qGamma{q}@{z} = \qPochhammer{q}{q}{\infty}(1-q)^{1-z}/\qPochhammer{q^{z}}{q}{\infty}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>QGAMMA(q, z) = QPochhammer(q, q, infinity)*(1 - q)^(1 - z)/QPochhammer((q)^(z), q, infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>QGamma[z,q] == QPochhammer[q, q, Infinity]*(1 - q)^(1 - z)/QPochhammer[(q)^(z), q, Infinity]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [56 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[QGamma[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994]], Times[Complex[-0.4701974928403924, -0.07292434984262404], Power[QPochhammer[Complex[0.6918839380246471, 0.3371668184918191], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1], QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[QGamma[Complex[-0.4999999999999998, 0.8660254037844387], Complex[0.8660254037844387, 0.49999999999999994]], Times[Complex[-0.021172596861766507, 0.11798586945608598], Power[QPochhammer[Complex[0.6137803977754971, -0.16446196191399762], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]], -1], QPochhammer[Complex[0.8660254037844387, 0.49999999999999994], Complex[0.8660254037844387, 0.49999999999999994], DirectedInfinity[1]]]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.18.E5 5.18.E5] || [[Item:Q2192|<math>\qGamma{q}@{1} = \qGamma{q}@{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\qGamma{q}@{1} = \qGamma{q}@{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>QGAMMA(q, 1) = QGAMMA(q, 2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>QGamma[1,q] == QGamma[2,q]</syntaxhighlight> || Error || Successful || - || Successful [Tested: 10]
|-
| [https://dlmf.nist.gov/5.18.E5 5.18.E5] || [[Item:Q2192|<math>\qGamma{q}@{2} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\qGamma{q}@{2} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>QGAMMA(q, 2) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>QGamma[2,q] == 1</syntaxhighlight> || Error || Successful || - || Successful [Tested: 10]
|-
| [https://dlmf.nist.gov/5.18.E6 5.18.E6] || [[Item:Q2193|<math>\qfactorial{n}{q} = \qGamma{q}@{n+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\qfactorial{n}{q} = \qGamma{q}@{n+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>QFactorial(n, q) = QGAMMA(q, n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>QFactorial[n,q] == QGamma[n + 1,q]</syntaxhighlight> || Error || Successful || - || Successful [Tested: 30]
|-
| [https://dlmf.nist.gov/5.18.E7 5.18.E7] || [[Item:Q2194|<math>\qGamma{q}@{z+1} = \frac{1-q^{z}}{1-q}\qGamma{q}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\qGamma{q}@{z+1} = \frac{1-q^{z}}{1-q}\qGamma{q}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>QGAMMA(q, z + 1) = (1 - (q)^(z))/(1 - q)*QGAMMA(q, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>QGamma[z + 1,q] == Divide[1 - (q)^(z),1 - q]*QGamma[z,q]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [17 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.18.E8 5.18.E8] || [[Item:Q2195|<math>\qGamma{q}@{x} < \qGamma{r}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\qGamma{q}@{x} < \qGamma{r}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>QGAMMA(q, x) < QGAMMA(r, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>QGamma[x,q] < QGamma[x,r]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [174 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Less[QGamma[1.5, Complex[0.8660254037844387, 0.49999999999999994]], Complex[0.4591522571856908, -0.3749002120921232]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[r, -1.5], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Less[QGamma[0.5, Complex[0.8660254037844387, 0.49999999999999994]], Complex[4.854857756142472*^-6, 0.9372445842681697]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[r, -1.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.18.E9 5.18.E9] || [[Item:Q2196|<math>\qGamma{q}@{x} > \qGamma{r}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\qGamma{q}@{x} > \qGamma{r}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>QGAMMA(q, x) > QGAMMA(r, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>QGamma[x,q] > QGamma[x,r]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [174 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Greater[QGamma[1.5, Complex[0.8660254037844387, 0.49999999999999994]], Complex[0.4591522571856908, -0.3749002120921232]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[r, -1.5], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Greater[QGamma[0.5, Complex[0.8660254037844387, 0.49999999999999994]], Complex[4.854857756142472*^-6, 0.9372445842681697]]
Test Values: {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[r, -1.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.18.E10 5.18.E10] || [[Item:Q2197|<math>\lim_{q\to 1-}\qGamma{q}@{z} = \EulerGamma@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\lim_{q\to 1-}\qGamma{q}@{z} = \EulerGamma@{z}</syntaxhighlight> || <math>\realpart@@{z} > 0</math> || <syntaxhighlight lang=mathematica>limit(QGAMMA(q, z), q = 1, left) = GAMMA(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Limit[QGamma[z,q], q -> 1, Direction -> "FromBelow", GenerateConditions->None] == Gamma[z]</syntaxhighlight> || Failure || Aborted || Error || Skipped - Because timed out
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.19#Ex1 5.19#Ex1] || [[Item:Q2200|<math>S = \sum_{k=0}^{\infty}a_{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>S = \sum_{k=0}^{\infty}a_{k}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">S = sum(a[k], k = 0..infinity)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">S == Sum[Subscript[a, k], {k, 0, Infinity}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.19#Ex2 5.19#Ex2] || [[Item:Q2201|<math>a_{k} = \frac{k}{(3k+2)(2k+1)(k+1)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{k} = \frac{k}{(3k+2)(2k+1)(k+1)}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[k] = (k)/((3*k + 2)*(2*k + 1)*(k + 1))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, k] == Divide[k,(3*k + 2)*(2*k + 1)*(k + 1)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/5.19.E2 5.19.E2] || [[Item:Q2202|<math>a_{k} = \frac{2}{k+\frac{2}{3}}-\frac{1}{k+\frac{1}{2}}-\frac{1}{k+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{k} = \frac{2}{k+\frac{2}{3}}-\frac{1}{k+\frac{1}{2}}-\frac{1}{k+1}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[k] = (2)/(k +(2)/(3))-(1)/(k +(1)/(2))-(1)/(k + 1)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, k] == Divide[2,k +Divide[2,3]]-Divide[1,k +Divide[1,2]]-Divide[1,k + 1]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-
| [https://dlmf.nist.gov/5.19.E3 5.19.E3] || [[Item:Q2203|<math>S = \digamma@{\tfrac{1}{2}}-2\digamma@{\tfrac{2}{3}}-\EulerConstant</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>S = \digamma@{\tfrac{1}{2}}-2\digamma@{\tfrac{2}{3}}-\EulerConstant</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>S = Psi((1)/(2))- 2*Psi((2)/(3))- gamma</syntaxhighlight> || <syntaxhighlight lang=mathematica>S == PolyGamma[Divide[1,2]]- 2*PolyGamma[Divide[2,3]]- EulerGamma</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .7702822630+.5000000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5957431410+.8660254040*I
Test Values: {S = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .4042568590-.8660254040*I
Test Values: {S = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.9617685450-.5000000000*I
Test Values: {S = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.7702822631342183, 0.49999999999999994]
Test Values: {Rule[S, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.5957431406502202, 0.8660254037844387]
Test Values: {Rule[S, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.19.E3 5.19.E3] || [[Item:Q2203|<math>\digamma@{\tfrac{1}{2}}-2\digamma@{\tfrac{2}{3}}-\EulerConstant = 3\ln@@{3}-2\ln@@{2}-\tfrac{1}{3}\pi\sqrt{3}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\digamma@{\tfrac{1}{2}}-2\digamma@{\tfrac{2}{3}}-\EulerConstant = 3\ln@@{3}-2\ln@@{2}-\tfrac{1}{3}\pi\sqrt{3}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Psi((1)/(2))- 2*Psi((2)/(3))- gamma = 3*ln(3)- 2*ln(2)-(1)/(3)*Pi*sqrt(3)</syntaxhighlight> || <syntaxhighlight lang=mathematica>PolyGamma[Divide[1,2]]- 2*PolyGamma[Divide[2,3]]- EulerGamma == 3*Log[3]- 2*Log[2]-Divide[1,3]*Pi*Sqrt[3]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 1]
|-
| [https://dlmf.nist.gov/5.19#Ex3 5.19#Ex3] || [[Item:Q2204|<math>V = \frac{\pi^{\frac{1}{2}n}r^{n}}{\EulerGamma@{\frac{1}{2}n+1}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>V = \frac{\pi^{\frac{1}{2}n}r^{n}}{\EulerGamma@{\frac{1}{2}n+1}}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2}n+1)} > 0</math> || <syntaxhighlight lang=mathematica>V = ((Pi)^((1)/(2)*n)* (r)^(n))/(GAMMA((1)/(2)*n + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>V == Divide[(Pi)^(Divide[1,2]*n)* (r)^(n),Gamma[Divide[1,2]*n + 1]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.866025403+.5000000000*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -6.202558068+.5000000000*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 15.00319234+.5000000000*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -2.133974595+.5000000000*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = 1.5, n = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[3.866025403784439, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[r, -1.5], Rule[V, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-6.202558066792596, 0.49999999999999994]
Test Values: {Rule[n, 2], Rule[r, -1.5], Rule[V, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.19#Ex4 5.19#Ex4] || [[Item:Q2205|<math>S = \frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\EulerGamma@{\frac{1}{2}n}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>S = \frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\EulerGamma@{\frac{1}{2}n}}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2}n)} > 0</math> || <syntaxhighlight lang=mathematica>S = (2*(Pi)^((1)/(2)*n)* (r)^(n - 1))/(GAMMA((1)/(2)*n))</syntaxhighlight> || <syntaxhighlight lang=mathematica>S == Divide[2*(Pi)^(Divide[1,2]*n)* (r)^(n - 1),Gamma[Divide[1,2]*n]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [174 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.133974596+.5000000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 10.29080337+.5000000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -27.40830850+.5000000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.133974596+.5000000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, r = 1.5, n = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [174 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.1339745962155612, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[r, -1.5], Rule[S, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[10.290803364553819, 0.49999999999999994]
Test Values: {Rule[n, 2], Rule[r, -1.5], Rule[S, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.19#Ex4 5.19#Ex4] || [[Item:Q2205|<math>\frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\EulerGamma@{\frac{1}{2}n}} = \frac{n}{r}V</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\EulerGamma@{\frac{1}{2}n}} = \frac{n}{r}V</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2}n)} > 0</math> || <syntaxhighlight lang=mathematica>(2*(Pi)^((1)/(2)*n)* (r)^(n - 1))/(GAMMA((1)/(2)*n)) = (n)/(r)*V</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[2*(Pi)^(Divide[1,2]*n)* (r)^(n - 1),Gamma[Divide[1,2]*n]] == Divide[n,r]*V</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.577350269+.3333333334*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -8.270077424+.6666666665*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 30.00638471+1.000000000*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = -1.5, n = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.422649731-.3333333334*I
Test Values: {V = 1/2*3^(1/2)+1/2*I, r = 1.5, n = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [180 / 180]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[2.5773502691896253, 0.33333333333333326]
Test Values: {Rule[n, 1], Rule[r, -1.5], Rule[V, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-8.270077422390127, 0.6666666666666665]
Test Values: {Rule[n, 2], Rule[r, -1.5], Rule[V, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-
| [https://dlmf.nist.gov/5.20.E1 5.20.E1] || [[Item:Q2206|<math>W = \frac{1}{2}\sum_{\ell=1}^{n}x_{\ell}^{2}-\sum_{1\leq\ell<j\leq n}\ln@@{|x_{\ell}-x_{j}|}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>W = \frac{1}{2}\sum_{\ell=1}^{n}x_{\ell}^{2}-\sum_{1\leq\ell<j\leq n}\ln@@{|x_{\ell}-x_{j}|}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>W = (1)/(2)*sum((x[ell])^(2), ell = 1..n)- sum(sum(ln(abs(x[ell]- x[j])), j = ell + 1..n), ell = 1..j - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>W == Divide[1,2]*Sum[(Subscript[x, \[ScriptL]])^(2), {\[ScriptL], 1, n}, GenerateConditions->None]- Sum[Sum[Log[Abs[Subscript[x, \[ScriptL]]- Subscript[x, j]]], {j, [ScriptL] + 1, n}, GenerateConditions->None], {\[ScriptL], 1, j - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || Error
|-
| [https://dlmf.nist.gov/5.20.E4 5.20.E4] || [[Item:Q2209|<math>W = -\sum_{1\leq\ell<j\leq n}\ln@@{|e^{i\theta_{\ell}}-e^{i\theta_{j}}|}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>W = -\sum_{1\leq\ell<j\leq n}\ln@@{|e^{i\theta_{\ell}}-e^{i\theta_{j}}|}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>W = - sum(sum(ln(abs(exp(I*theta[ell])- exp(I*theta[j]))), j = ell + 1..n), ell = 1..j - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>W == - Sum[Sum[Log[Abs[Exp[I*Subscript[\[Theta], \[ScriptL]]]- Exp[I*Subscript[\[Theta], j]]]], {j, [ScriptL] + 1, n}, GenerateConditions->None], {\[ScriptL], 1, j - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Error || Error
|}
</div>
</div>

Latest revision as of 16:41, 25 May 2021

Notation
5.1 Special Notation
Properties
5.2 Definitions
5.3 Graphics
5.4 Special Values and Extrema
5.5 Functional Relations
5.6 Inequalities
5.7 Series Expansions
5.8 Infinite Products
5.9 Integral Representations
5.10 Continued Fractions
5.11 Asymptotic Expansions
5.12 Beta Function
5.13 Integrals
5.14 Multidimensional Integrals
5.15 Polygamma Functions
5.16 Sums
5.17 Barnes’ Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BarnesG} -Function (Double Gamma Function)
5.18 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle q} -Gamma and Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle q} -Beta Functions
Applications
5.19 Mathematical Applications
5.20 Physical Applications
Computation
5.21 Methods of Computation
5.22 Tables
5.23 Approximations
5.24 Software
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