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variants of this functions
PolyLog




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > PolyLog[nu,z] > Identities > Functional identities > Relations of special kind





http://functions.wolfram.com/10.08.17.0056.01









  


  








Input Form





f[z w] == f[z] + f[w] - f[(z (1 - w))/(1 - z w)] - f[(w (1 - z))/(1 - z w)] && f[z] + f[1 - z] == Pi^2/6 /; f[z] == PolyLog[2, z] + (1/2) Log[z] Log[1 - z] && 0 < z < 1








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["f", "[", RowBox[List["z", " ", "w"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["f", "[", "z", "]"]], "+", RowBox[List["f", "[", "w", "]"]], "-", RowBox[List["f", "[", FractionBox[RowBox[List["z", RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]]]], RowBox[List["1", "-", RowBox[List["z", " ", "w"]]]]], "]"]], "-", RowBox[List["f", "[", FractionBox[RowBox[List["w", RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]]]], RowBox[List["1", "-", RowBox[List["z", " ", "w"]]]]], "]"]]]]]], "\[And]", RowBox[List[RowBox[List[RowBox[List["f", "[", "z", "]"]], "+", RowBox[List["f", "[", RowBox[List["1", "-", "z"]], "]"]]]], "\[Equal]", FractionBox[SuperscriptBox["\[Pi]", "2"], "6"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["f", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", "z"]], "]"]], "+", RowBox[List[FractionBox["1", "2"], RowBox[List["Log", "[", "z", "]"]], RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]]]]]], "\[And]", RowBox[List["0", "<", "z", "<", "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> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mi> w </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mi> w </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mn> 6 </mn> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> z </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <and /> <apply> <eq /> <apply> <ci> f </ci> <apply> <times /> <ci> z </ci> <ci> w </ci> </apply> </apply> <apply> <plus /> <apply> <ci> f </ci> <ci> w </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> f </ci> <apply> <times /> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <ci> w </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> f </ci> <apply> <times /> <ci> w </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <ci> w </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> f </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <eq /> <apply> <plus /> <apply> <ci> f </ci> <ci> z </ci> </apply> <apply> <ci> f </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> f </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <ln /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["f", "[", RowBox[List["z", " ", "w"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["f", "[", "z", "]"]], "+", RowBox[List["f", "[", "w", "]"]], "-", RowBox[List["f", "[", FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]]]], RowBox[List["1", "-", RowBox[List["z", " ", "w"]]]]], "]"]], "-", RowBox[List["f", "[", FractionBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]]]], RowBox[List["1", "-", RowBox[List["z", " ", "w"]]]]], "]"]]]]]], "&&", RowBox[List[RowBox[List[RowBox[List["f", "[", "z", "]"]], "+", RowBox[List["f", "[", RowBox[List["1", "-", "z"]], "]"]]]], "\[Equal]", FractionBox[SuperscriptBox["\[Pi]", "2"], "6"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["f", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", "z"]], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Log", "[", "z", "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]]]]]], "&&", RowBox[List["0", "<", "z", "<", "1"]]]]]]]]


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





2001-10-29