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http://functions.wolfram.com/10.07.16.0003.01
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PolyLog[2, z w] == PolyLog[2, w] + PolyLog[2, z] -
PolyLog[2, (w (1 - z))/(1 - w z)] - PolyLog[2, (z (1 - w))/(1 - w z)] +
(1/2) (Log[1 - w] Log[w] + Log[1 - z] Log[z] - Log[w z] Log[1 - w z] -
Log[(1 - w)/(1 - w z)] Log[(w (1 - z))/(1 - w z)] -
Log[(1 - z)/(1 - w z)] Log[(z (1 - w))/(1 - w z)]) /;
(Abs[z] < 1 && 0 < w < 1) || (Abs[w] < 1 && 0 < z < 1) ||
((Element[z, Reals] && Element[w, Reals] && (0 < z < 1 && w < 1)) ||
(0 < w < 1 && z < 1) || (z > 1 && w > 1))
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["z", " ", "w"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", "w"]], "]"]], "+", RowBox[List["PolyLog", "[", RowBox[List["2", ",", "z"]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]]]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]]]], "]"]], "-", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List["z", RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]]]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]]]], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", "w"]], "]"]], " ", RowBox[List["Log", "[", "w", "]"]]]], "+", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", RowBox[List["w", " ", "z"]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]], "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List["1", "-", "w"]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]], "]"]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]]]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]], "]"]]]], "-", RowBox[List[RowBox[List["Log", "[", FractionBox[RowBox[List["1", "-", "z"]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]], "]"]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["z", RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]]]], RowBox[List["1", "-", RowBox[List["w", " ", "z"]]]]], "]"]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]], "\[And]", RowBox[List["0", "<", "w", "<", "1"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", "1"]], "\[And]", RowBox[List["0", "<", "z", "<", "1"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "\[And]", RowBox[List["w", "\[Element]", "Reals"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["0", "<", "z", "<", "1"]], "\[And]", RowBox[List["w", "<", "1"]]]], ")"]]]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["0", "<", "w", "<", "1"]], "\[And]", RowBox[List["z", "<", "1"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["z", ">", "1"]], "\[And]", RowBox[List["w", ">", "1"]]]], ")"]]]], ")"]]]]]]]]
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<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> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </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> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> ⁢ </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> w </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mfrac> <mrow> <mi> w </mi> <mo> ⁢ </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> w </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> w </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> ⁢ </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> w </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> w </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> w </mi> <mo> ⁢ </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> w </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> w </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> w </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mn> 0 </mn> <mo> < </mo> <mi> w </mi> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∨ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> w </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mn> 0 </mn> <mo> < </mo> <mi> z </mi> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∨ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> z </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> w </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 0 </mn> <mo> < </mo> <mi> z </mi> <mo> < </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> w </mi> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 0 </mn> <mo> < </mo> <mi> w </mi> <mo> < </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∨ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> > </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> w </mi> <mo> > </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <ci> z </ci> <ci> w </ci> </apply> </apply> <apply> <plus /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <ci> w </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <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> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <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> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <ln /> <ci> w </ci> </apply> </apply> <apply> <times /> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <apply> <times /> <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> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ln /> <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> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <apply> <times /> <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> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ln /> <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> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <apply> <times /> <ci> w </ci> <ci> z </ci> </apply> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> w </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <ci> w </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <in /> <ci> z </ci> <reals /> </apply> <apply> <in /> <ci> w </ci> <reals /> </apply> <apply> <and /> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <ci> w </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> w </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <ci> w </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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