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variants of this functions
PolyLog






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > PolyLog[nu,z] > Integral representations > Contour integral representations





http://functions.wolfram.com/10.08.07.0008.02









  


  










Input Form





PolyLog[\[Nu], z] == (1/(2 Pi I)) Integrate[(Gamma[t] Gamma[-t] (-t)^(1 - \[Nu]))/(-z)^t, {t, \[Gamma] - I Infinity, \[Gamma] + I Infinity}] /; -1 < \[Gamma] < 0 && z != 0 && (Arg[-z] < Pi || Re[\[Nu]] > 1)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[RowBox[List["Gamma", "[", "t", "]"]], RowBox[List["Gamma", "[", RowBox[List["-", "t"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "t"]], ")"]], RowBox[List["1", "-", "\[Nu]"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "t"]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "<", "\[Gamma]", "<", "0"]], "\[And]", RowBox[List["z", "!=", "0"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Arg", "[", RowBox[List["-", "z"]], "]"]], "<", "\[Pi]"]], "\[Or]", RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", "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> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mrow> <mi> &#947; </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> <mo> + </mo> <mi> &#947; </mi> </mrow> </msubsup> <mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> t </mi> </mrow> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &lt; </mo> <mi> &#947; </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mi> &#960; </mi> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PolyLog </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <apply> <plus /> <ci> &#947; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <infinity /> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <plus /> <apply> <times /> <imaginaryi /> <infinity /> </apply> <ci> &#947; </ci> </apply> </uplimit> <apply> <times /> <apply> <ci> Gamma </ci> <ci> t </ci> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> &#947; </ci> <cn type='integer'> 0 </cn> </apply> <apply> <neq /> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <or /> <apply> <lt /> <apply> <arg /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <pi /> </apply> <apply> <gt /> <apply> <real /> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "t", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "t"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "t"]], ")"]], RowBox[List["1", "-", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "t"]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "<", "\[Gamma]", "<", "0"]], "&&", RowBox[List["z", "\[NotEqual]", "0"]], "&&", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Arg", "[", RowBox[List["-", "z"]], "]"]], "<", "\[Pi]"]], "||", RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", "1"]]]], ")"]]]]]]]]]]










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





2001-10-29