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variants of this functions
Zeta




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s] > Differentiation > Low-order differentiation > Derivatives at other points





http://functions.wolfram.com/10.01.20.0019.01









  


  








Input Form





Derivative[1][Zeta][-5] == 137/15120 + Log[Glaisher]/6 + (1/24) Log[2 Pi] - 120 PolyGamma[-7, 1] + 60 PolyGamma[-6, 1] - 10 PolyGamma[-5, 1]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "5"]], "]"]], "\[Equal]", RowBox[List[FractionBox["137", "15120"], "+", FractionBox[RowBox[List["Log", "[", "Glaisher", "]"]], "6"], "+", RowBox[List[FractionBox["1", "24"], " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], "-", RowBox[List["120", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "7"]], ",", "1"]], "]"]]]], "+", RowBox[List["60", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "6"]], ",", "1"]], "]"]]]], "-", RowBox[List["10", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "5"]], ",", "1"]], "]"]]]]]]]]]]








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> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 5 </mn> </mrow> <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> 6 </mn> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 24 </mn> </mfrac> <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> <mrow> <mn> 120 </mn> <mo> &#8290; </mo> <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> <mn> 1 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 60 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 10 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mn> 137 </mn> <mn> 15120 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -5 </cn> </apply> <cn type='integer'> -5 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ln /> <ci> Glaisher </ci> </apply> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 24 </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'> 120 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> -7 </cn> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 60 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> -6 </cn> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> -5 </cn> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='rational'> 137 <sep /> 15120 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "5"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["137", "15120"], "+", FractionBox[RowBox[List["Log", "[", "Glaisher", "]"]], "6"], "+", RowBox[List[FractionBox["1", "24"], " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], "-", RowBox[List["120", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "7"]], ",", "1"]], "]"]]]], "+", RowBox[List["60", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "6"]], ",", "1"]], "]"]]]], "-", RowBox[List["10", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "5"]], ",", "1"]], "]"]]]]]]]]]]


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





2007-05-02