Gamma Function - 5.15 Polygamma Functions
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 5.15.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma'@{z} = \sum_{k=0}^{\infty}\frac{1}{(k+z)^{2}}}
\digamma'@{z} = \sum_{k=0}^{\infty}\frac{1}{(k+z)^{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | diff( Psi(z), z$(1) ) = sum((1)/((k + z)^(2)), k = 0..infinity)
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D[PolyGamma[z], {z, 1}] == Sum[Divide[1,(k + z)^(2)], {k, 0, Infinity}, GenerateConditions->None]
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Failure | Successful | Successful [Tested: 0] | Successful [Tested: 1] |
| 5.15.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \polygamma{n}@{1} = (-1)^{n+1}n!\Riemannzeta@{n+1}}
\polygamma{n}@{1} = (-1)^{n+1}n!\Riemannzeta@{n+1} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Psi(n, 1) = (- 1)^(n + 1)* factorial(n)*Zeta(n + 1)
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PolyGamma[n, 1] == (- 1)^(n + 1)* (n)!*Zeta[n + 1]
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Failure | Failure | Successful [Tested: 3] | Successful [Tested: 3] |
| 5.15.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \polygamma{n}@{\tfrac{1}{2}} = (-1)^{n+1}n!(2^{n+1}-1)\Riemannzeta@{n+1}}
\polygamma{n}@{\tfrac{1}{2}} = (-1)^{n+1}n!(2^{n+1}-1)\Riemannzeta@{n+1} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Psi(n, (1)/(2)) = (- 1)^(n + 1)* factorial(n)*((2)^(n + 1)- 1)*Zeta(n + 1)
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PolyGamma[n, Divide[1,2]] == (- 1)^(n + 1)* (n)!*((2)^(n + 1)- 1)*Zeta[n + 1]
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Failure | Failure | Successful [Tested: 3] | Successful [Tested: 3] |
| 5.15.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma'@{n-\tfrac{1}{2}} = \tfrac{1}{2}\pi^{2}-4\sum_{k=1}^{n-1}\frac{1}{(2k-1)^{2}}}
\digamma'@{n-\tfrac{1}{2}} = \tfrac{1}{2}\pi^{2}-4\sum_{k=1}^{n-1}\frac{1}{(2k-1)^{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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)
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(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]
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Successful | Successful | - | Successful [Tested: 3] |
| 5.15.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma^{(n)}@{z+1} = \digamma^{(n)}@{z}+(-1)^{n}n!z^{-n-1}}
\digamma^{(n)}@{z+1} = \digamma^{(n)}@{z}+(-1)^{n}n!z^{-n-1} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | subs( temp=z + 1, diff( Psi(temp), temp$(n) ) ) = diff( Psi(z), z$(n) )+(- 1)^(n)* factorial(n)*(z)^(- n - 1)
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(D[PolyGamma[temp], {temp, n}]/.temp-> z + 1) == D[PolyGamma[z], {z, n}]+(- 1)^(n)* (n)!*(z)^(- n - 1)
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Failure | Failure | Successful [Tested: 21] | Successful [Tested: 21] |
| 5.15.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma^{(n)}@{1-z}+(-1)^{n-1}\digamma^{(n)}@{z} = (-1)^{n}\pi\deriv[n]{}{z}\cot@{\pi z}}
\digamma^{(n)}@{1-z}+(-1)^{n-1}\digamma^{(n)}@{z} = (-1)^{n}\pi\deriv[n]{}{z}\cot@{\pi z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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$(n)])
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(D[PolyGamma[temp], {temp, n}]/.temp-> 1 - z)+(- 1)^(n - 1)* D[PolyGamma[z], {z, n}] == (- 1)^(n)* Pi*D[Cot[Pi*z], {z, n}]
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Failure | Failure | Error | Failed [21 / 21]
Result: Plus[Complex[-1.111486978443634, 1.4242909397222407], Times[Complex[-1.1253971041044755, 1.3474673991212198], Inactive[Sum][Times[Power[-0.5, K[1.0]], Power[Complex[2.0570132833277626, -0.06826349589921218], K[1.0]], Factorial[K[1.0]], StirlingS2[1.0, K[1.0]]]
Test Values: {K[1.0], 0.0, 1.0}]]], {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Plus[Complex[9.936030880873925, 6.770945349247037], Times[Complex[-8.466387364061939, -7.071078549251696], Inactive[Sum][Times[Power[-0.5, K[1.0]], Power[Complex[2.0570132833277626, -0.06826349589921218], K[1.0]], Factorial[K[1.0]], StirlingS2[2.0, K[1.0]]]
Test Values: {K[1.0], 0.0, 2.0}]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |
| 5.15.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma^{(n)}@{mz} = \frac{1}{m^{n+1}}\sum_{k=0}^{m-1}\digamma^{(n)}@{z+\frac{k}{m}}}
\digamma^{(n)}@{mz} = \frac{1}{m^{n+1}}\sum_{k=0}^{m-1}\digamma^{(n)}@{z+\frac{k}{m}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | diff( Psi(m*z), m*z$(n) ) = (1)/((m)^(n + 1))*sum(subs( temp=z +(k)/(m), diff( Psi(temp), temp$(n) ) ), k = 0..m - 1)
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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]
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Error | Failure | - | Failed [63 / 63]
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]]]}
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]]]}
... skip entries to safe data |