View source for 5.2

{{DISPLAYTITLE:Elementary Functions - 5.2 Definitions}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! 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
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| [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]
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| [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]
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| [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]
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| [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]
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| [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]
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| [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]
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| [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]
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