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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/06.15.03.0037.01









  


  










Input Form





PolyGamma[-7, z] == (1/(1814400 Pi^6)) (Pi^6 z (-10 + 7 z^2 (37 + 9 z^2 (-49 + 240 Log[Glaisher] + z (49 - 14 z + 20 Log[2 Pi])))) + 14175 Zeta[7] - 630 Pi^2 z (4 Pi^4 z^5 Log[z] - 15 Pi^2 z^3 Zeta[3] + 45 z Zeta[5] + 8 Pi^4 (3 Derivative[1][Zeta][-5] + 10 z^2 Derivative[1][Zeta][-3])) + 2520 Pi^6 Derivative[1, 0][Zeta][-6, 1 + z])










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> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 1814400 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 6 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#960; </mi> <mn> 6 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 240 </mn> <mo> &#8290; </mo> <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> </mrow> <mo> + </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 14 </mn> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 20 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 49 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 49 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 37 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 10 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 14175 </mn> <mo> &#8290; </mo> <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> <mo> - </mo> <mrow> <mn> 630 </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> 4 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 4 </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> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </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; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 45 </mn> <mo> &#8290; </mo> <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> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 10 </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> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </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> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2520 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 6 </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> 6 </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'> -7 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 1814400 </cn> <apply> <power /> <pi /> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 6 </cn> </apply> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 240 </cn> <apply> <ln /> <ci> Glaisher </ci> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -14 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 20 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> <cn type='integer'> 49 </cn> </apply> </apply> <cn type='integer'> -49 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 37 </cn> </apply> </apply> <cn type='integer'> -10 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 14175 </cn> <apply> <ci> Zeta </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 630 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <ln /> <ci> z </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <ci> Zeta </ci> <cn type='integer'> 5 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 10 </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'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -5 </cn> </apply> <cn type='integer'> -5 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2520 </cn> <apply> <power /> <pi /> <cn type='integer'> 6 </cn> </apply> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> </list> <ci> Zeta </ci> </apply> <cn type='integer'> -6 </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["-", "7"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["\[Pi]", "6"], " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "10"]], "+", RowBox[List["7", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["37", "+", RowBox[List["9", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "49"]], "+", RowBox[List["240", " ", RowBox[List["Log", "[", "Glaisher", "]"]]]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["49", "-", RowBox[List["14", " ", "z"]], "+", RowBox[List["20", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["14175", " ", RowBox[List["Zeta", "[", "7", "]"]]]], "-", RowBox[List["630", " ", SuperscriptBox["\[Pi]", "2"], " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["\[Pi]", "4"], " ", SuperscriptBox["z", "5"], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["15", " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "3"], " ", RowBox[List["Zeta", "[", "3", "]"]]]], "+", RowBox[List["45", " ", "z", " ", RowBox[List["Zeta", "[", "5", "]"]]]], "+", RowBox[List["8", " ", SuperscriptBox["\[Pi]", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "5"]], "]"]]]], "+", RowBox[List["10", " ", SuperscriptBox["z", "2"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "3"]], "]"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["2520", " ", SuperscriptBox["\[Pi]", "6"], " ", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["-", "6"]], ",", RowBox[List["1", "+", "z"]]]], "]"]]]]]], RowBox[List["1814400", " ", SuperscriptBox["\[Pi]", "6"]]]]]]]]










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





2007-05-02





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