Conjugate[LogIntegral[x + I y]] == (1/2) (LogIntegral[x - x Sqrt[-(y^2/x^2)]] + LogIntegral[x + x Sqrt[-(y^2/x^2)]]) - I (x/(2 y)) Sqrt[-(y^2/x^2)] (LogIntegral[x - x Sqrt[-(y^2/x^2)]] - LogIntegral[x + x Sqrt[-(y^2/x^2)]])

Cell[BoxData[RowBox[List[RowBox[List["Conjugate", "[", RowBox[List["LogIntegral", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["LogIntegral", "[", RowBox[List["x", "-", RowBox[List["x", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["y", "2"], SuperscriptBox["x", "2"]]]]]]]]], "]"]], "+", RowBox[List["LogIntegral", "[", RowBox[List["x", "+", RowBox[List["x", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["y", "2"], SuperscriptBox["x", "2"]]]]]]]]], "]"]]]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", FractionBox["x", RowBox[List["2", "y"]]], SqrtBox[RowBox[List["-", FractionBox[RowBox[List[" ", SuperscriptBox["y", "2"]]], SuperscriptBox["x", "2"]]]]], RowBox[List["(", RowBox[List[RowBox[List["LogIntegral", "[", RowBox[List["x", "-", RowBox[List["x", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["y", "2"], SuperscriptBox["x", "2"]]]]]]]]], "]"]], "-", RowBox[List["LogIntegral", "[", RowBox[List["x", "+", RowBox[List["x", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["y", "2"], SuperscriptBox["x", "2"]]]]]]]]], "]"]]]], ")"]]]]]]]]]]

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

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

2001-10-29