Limit[LerchPhiClassical[x + I \[Epsilon], s, a], \[Epsilon] -> Plus[0]] == LerchPhiClassical[x, s, a] + ((2 I Pi)/(x^a Gamma[s])) Log[x]^(s - 1) /; x > 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> <munder> <mi> lim </mi> <mrow> <mi> &#1013; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <semantics> <mrow> <mover> <mi> &#934; </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#1013; </mi> </mrow> </mrow> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[OverscriptBox[&quot;\[CapitalPhi]&quot;, &quot;^&quot;], &quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;x&quot;, &quot;+&quot;, RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;\[Epsilon]&quot;]]]], Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;s&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;a&quot;, Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2, $CellContext`e3], LerchPhi[$CellContext`e1, $CellContext`e2, $CellContext`e3]]]] </annotation> </semantics> </mrow> <mo> &#10869; </mo> <mrow> <semantics> <mrow> <mover> <mi> &#934; </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[OverscriptBox[&quot;\[CapitalPhi]&quot;, &quot;^&quot;], &quot;(&quot;, RowBox[List[TagBox[&quot;x&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;s&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;a&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> + </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> x </mi> <mrow> <mo> - </mo> <mi> a </mi> </mrow> </msup> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> x </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> &#1013; </ci> </bvar> <condition> <apply> <tendsto /> <ci> &#1013; </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <ci> LerchPhi </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> &#1013; </ci> </apply> </apply> <ci> s </ci> <ci> a </ci> </apply> </apply> <apply> <plus /> <apply> <ci> LerchPhi </ci> <ci> x </ci> <ci> s </ci> <ci> a </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <power /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ln /> <ci> x </ci> </apply> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29