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variants of this functions
PolyLog






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > PolyLog[nu,z] > Representations through equivalent functions > With related functions





http://functions.wolfram.com/10.08.27.0003.01









  


  










Input Form





PolyLog[\[Nu], z] == (2 Pi)^(\[Nu] - 1) I Gamma[1 - \[Nu]] (Zeta[1 - \[Nu], (Log[-z] + Pi I)/(2 Pi I)]/E^((Pi I \[Nu])/2) - E^((Pi I \[Nu])/2) Zeta[1 - \[Nu], 1 - (Log[-z] + Pi I)/(2 Pi I)]) /; !IntervalMemberQ[Interval[{0, 1}], z]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", "\[Pi]"]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], "\[ImaginaryI]", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]], "2"]]]], RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]]], RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]], "2"]], RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]]], RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]]]]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]










MathML Form







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</mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> , </mo> <mfrac> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;\[Nu]&quot;]], Zeta, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[RowBox[List[&quot;log&quot;, &quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;z&quot;]], &quot;)&quot;]], &quot;+&quot;, RowBox[List[&quot;\[Pi]&quot;, &quot; 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</mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;\[Nu]&quot;]], Zeta, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, FractionBox[RowBox[List[RowBox[List[&quot;log&quot;, &quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;z&quot;]], &quot;)&quot;]], &quot;+&quot;, RowBox[List[&quot;\[Pi]&quot;, &quot; &quot;, &quot;\[ImaginaryI]&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;\[Pi]&quot;, &quot; &quot;, &quot;\[ImaginaryI]&quot;]]]]], Zeta, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <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> &#63449; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#960; 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&quot;, &quot;\[ImaginaryI]&quot;]]]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;\[Pi]&quot;, &quot; &quot;, &quot;\[ImaginaryI]&quot;]]], Zeta, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; 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</mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], " ", "\[ImaginaryI]", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]], ")"]]]]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]]]]]], "]"]]]]]], ")"]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]










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





2007-05-02





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