Gamma Function - 5.5 Functional Relations
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 5.5.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{z+1} = z\EulerGamma@{z}}
\EulerGamma@{z+1} = z\EulerGamma@{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(z+1)} > 0, \realpart@@{z} > 0} | GAMMA(z + 1) = z*GAMMA(z)
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Gamma[z + 1] == z*Gamma[z]
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Successful | Successful | - | Successful [Tested: 5] |
| 5.5.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{z+1} = \digamma@{z}+\frac{1}{z}}
\digamma@{z+1} = \digamma@{z}+\frac{1}{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Psi(z + 1) = Psi(z)+(1)/(z)
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PolyGamma[z + 1] == PolyGamma[z]+Divide[1,z]
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Successful | Successful | - | Successful [Tested: 7] |
| 5.5.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{z}\EulerGamma@{1-z} = \pi/\sin@{\pi z}}
\EulerGamma@{z}\EulerGamma@{1-z} = \pi/\sin@{\pi z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{z} > 0, \realpart@@{(1-z)} > 0} | GAMMA(z)*GAMMA(1 - z) = Pi/sin(Pi*z)
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Gamma[z]*Gamma[1 - z] == Pi/Sin[Pi*z]
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Successful | Successful | - | Successful [Tested: 1] |
| 5.5.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{z}-\digamma@{1-z} = -\pi/\tan@{\pi z}}
\digamma@{z}-\digamma@{1-z} = -\pi/\tan@{\pi z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Psi(z)- Psi(1 - z) = - Pi/tan(Pi*z)
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PolyGamma[z]- PolyGamma[1 - z] == - Pi/Tan[Pi*z]
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Successful | Successful | - | Successful [Tested: 1] |
| 5.5.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{2z} = \pi^{-1/2}2^{2z-1}\EulerGamma@{z}\EulerGamma@{z+\tfrac{1}{2}}}
\EulerGamma@{2z} = \pi^{-1/2}2^{2z-1}\EulerGamma@{z}\EulerGamma@{z+\tfrac{1}{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(2z)} > 0, \realpart@@{z} > 0, \realpart@@{(z+\tfrac{1}{2})} > 0} | GAMMA(2*z) = (Pi)^(- 1/2)* (2)^(2*z - 1)* GAMMA(z)*GAMMA(z +(1)/(2))
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Gamma[2*z] == (Pi)^(- 1/2)* (2)^(2*z - 1)* Gamma[z]*Gamma[z +Divide[1,2]]
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Successful | Successful | - | Successful [Tested: 5] |
| 5.5.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{nz} = (2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\EulerGamma@{z+\frac{k}{n}}}
\EulerGamma@{nz} = (2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\EulerGamma@{z+\frac{k}{n}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | GAMMA(n*z) = (2*Pi)^((1 - n)/2)* (n)^(n*z -(1/2))* product(GAMMA(z +(k)/(n)), k = 0..n - 1)
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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]
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Failure | Successful | Successful [Tested: 15] | Successful [Tested: 15] |
| 5.5.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \prod_{k=1}^{n-1}\EulerGamma@{\frac{k}{n}} = (2\pi)^{(n-1)/2}n^{-1/2}}
\prod_{k=1}^{n-1}\EulerGamma@{\frac{k}{n}} = (2\pi)^{(n-1)/2}n^{-1/2} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | product(GAMMA((k)/(n)), k = 1..n - 1) = (2*Pi)^((n - 1)/2)* (n)^(- 1/2)
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Product[Gamma[Divide[k,n]], {k, 1, n - 1}, GenerateConditions->None] == (2*Pi)^((n - 1)/2)* (n)^(- 1/2)
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Failure | Failure | Successful [Tested: 3] | Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[n, 1]}
Result: Indeterminate
Test Values: {Rule[n, 2]}
... skip entries to safe data |
| 5.5.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{2z} = \tfrac{1}{2}\left(\digamma@{z}+\digamma@{z+\tfrac{1}{2}}\right)+\ln@@{2}}
\digamma@{2z} = \tfrac{1}{2}\left(\digamma@{z}+\digamma@{z+\tfrac{1}{2}}\right)+\ln@@{2} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Psi(2*z) = (1)/(2)*(Psi(z)+ Psi(z +(1)/(2)))+ ln(2)
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PolyGamma[2*z] == Divide[1,2]*(PolyGamma[z]+ PolyGamma[z +Divide[1,2]])+ Log[2]
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Successful | Successful | - | Successful [Tested: 7] |
| 5.5.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{nz} = \frac{1}{n}\sum_{k=0}^{n-1}\digamma@{z+\frac{k}{n}}+\ln@@{n}}
\digamma@{nz} = \frac{1}{n}\sum_{k=0}^{n-1}\digamma@{z+\frac{k}{n}}+\ln@@{n} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Psi(n*z) = (1)/(n)*sum(Psi(z +(k)/(n)), k = 0..n - 1)+ ln(n)
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PolyGamma[n*z] == Divide[1,n]*Sum[PolyGamma[z +Divide[k,n]], {k, 0, n - 1}, GenerateConditions->None]+ Log[n]
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Failure | Successful | Successful [Tested: 21] | Successful [Tested: 21] |