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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 nu





http://functions.wolfram.com/06.15.03.0019.01









  


  










Input Form





PolyGamma[-9, 1] == -(16427/1524096000) + Log[Glaisher]/5040 + Log[2 Pi]/80640 + Zeta[3]/(5760 Pi^2) - Zeta[5]/(768 Pi^4) + Zeta[7]/(256 Pi^6) - Derivative[1][Zeta][-7]/5040 - (1/720) Derivative[1][Zeta][-5] - (1/720) Derivative[1][Zeta][-3]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "9"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["16427", "1524096000"]]], "+", FractionBox[RowBox[List["Log", "[", "Glaisher", "]"]], "5040"], "+", FractionBox[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "80640"], "+", FractionBox[RowBox[List["Zeta", "[", "3", "]"]], RowBox[List["5760", " ", SuperscriptBox["\[Pi]", "2"]]]], "-", FractionBox[RowBox[List["Zeta", "[", "5", "]"]], RowBox[List["768", " ", SuperscriptBox["\[Pi]", "4"]]]], "+", FractionBox[RowBox[List["Zeta", "[", "7", "]"]], RowBox[List["256", " ", SuperscriptBox["\[Pi]", "6"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "7"]], "]"]], "5040"], "-", RowBox[List[FractionBox["1", "720"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "5"]], "]"]]]], "-", RowBox[List[FractionBox["1", "720"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "3"]], "]"]]]]]]]]]]










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> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <semantics> <mi> A </mi> <annotation encoding='Mathematica'> TagBox[&quot;A&quot;, Function[List[], Glaisher]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 5040 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 80640 </mn> </mfrac> <mo> + </mo> <mfrac> <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> <mrow> <mn> 5760 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 7 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;7&quot;, Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mrow> <mn> 256 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 6 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 720 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 720 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 5040 </mn> </mfrac> <mo> - </mo> <mfrac> <mn> 16427 </mn> <mn> 1524096000 </mn> </mfrac> <mo> - </mo> <mfrac> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 5 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;5&quot;, Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mrow> <mn> 768 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> PolyGamma </ci> <cn type='integer'> -9 </cn> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ln /> <ci> Glaisher </ci> </apply> <apply> <power /> <cn type='integer'> 5040 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> <apply> <power /> <cn type='integer'> 80640 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 5760 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Zeta </ci> <cn type='integer'> 7 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <power /> <pi /> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 720 </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='rational'> 1 <sep /> 720 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -3 </cn> </apply> <cn type='integer'> -3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -7 </cn> </apply> <cn type='integer'> -7 </cn> </apply> <apply> <power /> <cn type='integer'> 5040 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 16427 <sep /> 1524096000 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Zeta </ci> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 768 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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"]], ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["16427", "1524096000"]]], "+", FractionBox[RowBox[List["Log", "[", "Glaisher", "]"]], "5040"], "+", FractionBox[RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], "80640"], "+", FractionBox[RowBox[List["Zeta", "[", "3", "]"]], RowBox[List["5760", " ", SuperscriptBox["\[Pi]", "2"]]]], "-", FractionBox[RowBox[List["Zeta", "[", "5", "]"]], RowBox[List["768", " ", SuperscriptBox["\[Pi]", "4"]]]], "+", FractionBox[RowBox[List["Zeta", "[", "7", "]"]], RowBox[List["256", " ", SuperscriptBox["\[Pi]", "6"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "7"]], "]"]], "5040"], "-", RowBox[List[FractionBox["1", "720"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "5"]], "]"]]]], "-", RowBox[List[FractionBox["1", "720"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "3"]], "]"]]]]]]]]]]










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





2007-05-02