Polygamma function: Specific values (formula 06.15.03.0020)

PolyGamma

$Mathematica Notation$

http://functions.wolfram.com/06.15.03.0020.01

Input Form

PolyGamma[-10, 1] == (1/(3048192000 Pi^8)) (75600 Pi^6 Zeta[3] - 793800 Pi^4 Zeta[5] + 3969000 Pi^2 Zeta[7] - 5953500 Zeta[9] + Pi^8 (-4457 + 75600 Log[Glaisher] + 4200 Log[2 Pi] - 302400 Derivative[1][Zeta][-7] - 1058400 Derivative[1][Zeta][-5] - 705600 Derivative[1][Zeta][-3]))

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 10 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 3048192000 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 8 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 75600 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 6 </mn> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["3", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mn> 793800 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 5 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["5", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mn> 3969000 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 7 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["7", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mn> 5953500 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 9 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["9", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <msup> <mi> π </mi> <mn> 8 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 75600 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mi> A </mi> <annotation encoding='Mathematica'> TagBox["A", Function[List[], Glaisher]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4200 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 302400 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 1058400 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 705600 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 4457 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> PolyGamma </ci> <cn type='integer'> -10 </cn> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 3048192000 </cn> <apply> <power /> <pi /> <cn type='integer'> 8 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 75600 </cn> <apply> <power /> <pi /> <cn type='integer'> 6 </cn> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 793800 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3969000 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5953500 </cn> <apply> <ci> Zeta </ci> <cn type='integer'> 9 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 8 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 75600 </cn> <apply> <ln /> <ci> Glaisher </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 4200 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 302400 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -7 </cn> </apply> <cn type='integer'> -7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1058400 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -5 </cn> </apply> <cn type='integer'> -5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 705600 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -3 </cn> </apply> <cn type='integer'> -3 </cn> </apply> </apply> </apply> <cn type='integer'> -4457 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02

PolyGamma[z]