LerchPhi[(1 - I)/2, 2, 1] == (1 - I) (Catalan + (5 I Pi^2)/96 - (Pi/8) Log[2] - (I/8) Log[2]^2)

Cell[BoxData[RowBox[List[RowBox[List["LerchPhi", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[ImaginaryI]"]], "2"], ",", "2", ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List["Catalan", "+", FractionBox[RowBox[List["5", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[Pi]", "2"]]], "96"], "-", RowBox[List[FractionBox["\[Pi]", "8"], RowBox[List["Log", "[", "2", "]"]]]], "-", RowBox[List[FractionBox["\[ImaginaryI]", "8"], SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "2"]]]]], ")"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mi> &#934; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[CapitalPhi]&quot;, &quot;(&quot;, RowBox[List[TagBox[FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;\[ImaginaryI]&quot;]], &quot;2&quot;], Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;2&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;1&quot;, Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2, $CellContext`e3], LerchPhi[$CellContext`e1, $CellContext`e2, $CellContext`e3]]]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#8520; </mi> <mn> 8 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mi> &#960; </mi> <mn> 8 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> <mn> 96 </mn> </mfrac> <mo> + </mo> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, Function[Catalan]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LerchPhi </ci> <apply> <times /> <cn type='complex-cartesian'> 1 <sep /> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='complex-cartesian'> 1 <sep /> -1 </cn> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 8 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 8 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <imaginaryi /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 96 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> Catalan </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LerchPhi", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[ImaginaryI]"]], "2"], ",", "2", ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List["Catalan", "+", FractionBox[RowBox[List["5", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[Pi]", "2"]]], "96"], "-", RowBox[List[FractionBox["1", "8"], " ", "\[Pi]", " ", RowBox[List["Log", "[", "2", "]"]]]], "-", RowBox[List[FractionBox["1", "8"], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "2"]]]]], ")"]]]]]]]]

2001-10-29