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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/10.08.27.0002.01









  


  










Input Form





PolyLog[\[Nu], z] == (2 Pi)^(\[Nu] - 1) I Gamma[1 - \[Nu]] (Zeta[1 - \[Nu], 1 + Floor[-(Arg[z]/(2 Pi))] + Log[z]/(2 Pi I)]/ E^((Pi I \[Nu])/2) - E^((Pi I \[Nu])/2) Zeta[1 - \[Nu], -Floor[-(Arg[z]/(2 Pi))] - Log[z]/(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]"]], ",", RowBox[List["1", "+", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "+", FractionBox[RowBox[List["Log", "[", "z", "]"]], 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[RowBox[List["-", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], "-", FractionBox[RowBox[List["Log", "[", "z", "]"]], 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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&quot;, &quot;\[Pi]&quot;]]]]], &quot;\[RightFloor]&quot;]]]], &quot;-&quot;, FractionBox[RowBox[List[&quot;log&quot;, &quot;(&quot;, &quot;z&quot;, &quot;)&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> </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]"]], ",", RowBox[List["1", "+", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "+", FractionBox[RowBox[List["Log", "[", "z", "]"]], 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[RowBox[List["-", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], "-", FractionBox[RowBox[List["Log", "[", "z", "]"]], 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