Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Zeta




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s] > Identities > Functional identities > Including derivatives of the function





http://functions.wolfram.com/10.01.17.0004.01









  


  








Input Form





Derivative[n][Zeta][1 - z] == (-1)^n Sum[Binomial[n, k] (E^(z ((-(1/2)) (I Pi) - Log[2 Pi])) ((-(1/2)) (I Pi) - Log[2 Pi])^(n - k) + E^(z ((I Pi)/2 - Log[2 Pi])) ((I Pi)/2 - Log[2 Pi])^(n - k)) D[Gamma[z] Zeta[z], {z, k}], {k, 0, n}] /; Element[n, Integers] && n >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", "n", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", "-", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List["(", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]]]], "-", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]]]], "-", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], ")"]], RowBox[List["n", "-", "k"]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"], "-", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"], "-", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], ")"]], RowBox[List["n", "-", "k"]]]]]]], ")"]], RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "k"]], "}"]]], RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "z", "]"]], " ", RowBox[List["Zeta", "[", "z", "]"]]]], ")"]]]]]], " ", ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> &#950; </mi> <semantics> <mrow> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;n&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;n&quot;, Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </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> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </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> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </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> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </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> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> k </mi> </msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <ci> n </ci> </list> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <pi /> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <pi /> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> k </ci> </degree> </bvar> <apply> <times /> <apply> <ci> Gamma </ci> <ci> z </ci> </apply> <apply> <ci> Zeta </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", "n", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", "-", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]]]], "-", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]]]], "-", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], ")"]], RowBox[List["n", "-", "k"]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"], "-", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"], "-", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], ")"]], RowBox[List["n", "-", "k"]]]]]]], ")"]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z", ",", "k"]], "}"]]]]], RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "z", "]"]], " ", RowBox[List["Zeta", "[", "z", "]"]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2002-12-18