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variants of this functions
PolyLog






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > PolyLog[2,z] > Complex characteristics > Real part





http://functions.wolfram.com/10.07.19.0001.01









  


  










Input Form





Re[PolyLog[2, x + I y]] == (-(1/4)) (Log[x^2 + y^2] (Log[1 - 2 x + x^2 + y^2] - Log[1 + (x^2 + y^2)/(-x + Sqrt[-y^2])] - Log[1 - (x^2 + y^2)/(x + Sqrt[-y^2])]) - 2 PolyLog[2, (x^2 + y^2)/(x - Sqrt[-y^2])] - 2 PolyLog[2, (x^2 + y^2)/(x + Sqrt[-y^2])])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Re", "[", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]]]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["2", " ", "x"]], "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], RowBox[List[RowBox[List["-", "x"]], "+", SqrtBox[RowBox[List["-", SuperscriptBox["y", "2"]]]]]]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], RowBox[List["x", "+", SqrtBox[RowBox[List["-", SuperscriptBox["y", "2"]]]]]]]]], "]"]]]], ")"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], RowBox[List["x", "-", SqrtBox[RowBox[List["-", SuperscriptBox["y", "2"]]]]]]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], RowBox[List["x", "+", SqrtBox[RowBox[List["-", SuperscriptBox["y", "2"]]]]]]]]], "]"]]]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <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> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <msqrt> <mrow> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> - </mo> <mi> x </mi> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> x </mi> <mo> + </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </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> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> x </mi> <mo> - </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </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> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> x </mi> <mo> + </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <real /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ln /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> </apply> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> x </ci> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> x </ci> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Re", "[", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "-", RowBox[List["2", " ", "x"]], "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], RowBox[List[RowBox[List["-", "x"]], "+", SqrtBox[RowBox[List["-", SuperscriptBox["y", "2"]]]]]]]]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], RowBox[List["x", "+", SqrtBox[RowBox[List["-", SuperscriptBox["y", "2"]]]]]]]]], "]"]]]], ")"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], RowBox[List["x", "-", SqrtBox[RowBox[List["-", SuperscriptBox["y", "2"]]]]]]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", FractionBox[RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], RowBox[List["x", "+", SqrtBox[RowBox[List["-", SuperscriptBox["y", "2"]]]]]]]]], "]"]]]]]], ")"]]]]]]]]










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





2001-10-29





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