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variants of this functions
PolyGamma






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > PolyGamma[nu,z] > Specific values > Specialized values > For fixed z





http://functions.wolfram.com/06.15.03.0039.01









  


  










Input Form





PolyGamma[-9, z] == (1/(3048192000 Pi^8)) (Pi^8 z (315 - 7300 z^2 + 31206 z^4 - 22830 z^8 + 180 z^6 (-761 + 3360 Log[Glaisher]) + 135 z^7 (761 + 280 Log[2 Pi])) - 5953500 Zeta[9] - 37800 Pi^2 z (2 Pi^6 z^7 Log[z] - 14 Pi^4 z^5 Zeta[3] + 105 Pi^2 z^3 Zeta[5] - 315 z Zeta[7] + 16 Pi^6 (Derivative[1][Zeta][-7] + 7 z^2 Derivative[1][Zeta][-5] + 7 z^4 Derivative[1][Zeta][-3])) + 75600 Pi^8 Derivative[1, 0][Zeta][-8, 1 + z])










Standard Form





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MathML Form







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</mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 9 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;9&quot;, Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mn> 37800 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 6 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 14 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;3&quot;, Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> &#8290; 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</mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 6 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 75600 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 8 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> &#950; </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;,&quot;, &quot;0&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> , </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </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'> -9 </cn> <ci> z </ci> </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 /> <apply> <power /> <pi /> <cn type='integer'> 8 </cn> </apply> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -22830 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 135 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 280 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> <cn type='integer'> 761 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 180 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3360 </cn> <apply> <ln /> <ci> Glaisher </ci> </apply> </apply> <cn type='integer'> -761 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 31206 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7300 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 315 </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 /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 37800 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> 6 </cn> </apply> <apply> <ln /> <ci> z </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 14 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 105 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 315 </cn> <apply> <ci> Zeta </ci> <cn type='integer'> 7 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <pi /> <cn type='integer'> 6 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -3 </cn> </apply> <cn type='integer'> -3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -5 </cn> </apply> <cn type='integer'> -5 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -7 </cn> </apply> <cn type='integer'> -7 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 75600 </cn> <apply> <power /> <pi /> <cn type='integer'> 8 </cn> </apply> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> </list> <ci> Zeta </ci> </apply> <cn type='integer'> -8 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "9"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["\[Pi]", "8"], " ", "z", " ", RowBox[List["(", RowBox[List["315", "-", RowBox[List["7300", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["31206", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["22830", " ", SuperscriptBox["z", "8"]]], "+", RowBox[List["180", " ", SuperscriptBox["z", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "761"]], "+", RowBox[List["3360", " ", RowBox[List["Log", "[", "Glaisher", "]"]]]]]], ")"]]]], "+", RowBox[List["135", " ", SuperscriptBox["z", "7"], " ", RowBox[List["(", RowBox[List["761", "+", RowBox[List["280", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]]]], ")"]]]]]], ")"]]]], "-", RowBox[List["5953500", " ", RowBox[List["Zeta", "[", "9", "]"]]]], "-", RowBox[List["37800", " ", SuperscriptBox["\[Pi]", "2"], " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["\[Pi]", "6"], " ", SuperscriptBox["z", "7"], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["14", " ", SuperscriptBox["\[Pi]", "4"], " ", SuperscriptBox["z", "5"], " ", RowBox[List["Zeta", "[", "3", "]"]]]], "+", RowBox[List["105", " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "3"], " ", RowBox[List["Zeta", "[", "5", "]"]]]], "-", RowBox[List["315", " ", "z", " ", RowBox[List["Zeta", "[", "7", "]"]]]], "+", RowBox[List["16", " ", SuperscriptBox["\[Pi]", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "7"]], "]"]], "+", RowBox[List["7", " ", SuperscriptBox["z", "2"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "5"]], "]"]]]], "+", RowBox[List["7", " ", SuperscriptBox["z", "4"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "3"]], "]"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["75600", " ", SuperscriptBox["\[Pi]", "8"], " ", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["-", "8"]], ",", RowBox[List["1", "+", "z"]]]], "]"]]]]]], RowBox[List["3048192000", " ", SuperscriptBox["\[Pi]", "8"]]]]]]]]










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





2007-05-02





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