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http://functions.wolfram.com/01.03.23.0013.01
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Sum[((-1)^k E^(k w))/(a^2 + k^2), {k, 1, Infinity}] ==
((I E^w)/(2 a)) (LerchPhi[-E^w, 1, 1 - I a] - LerchPhi[-E^w, 1,
1 + I a]) /; Re[w] <= 0
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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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> w </mi> </mrow> </msup> </mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> k </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mi> w </mi> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> ⅇ </mi> <mi> w </mi> </msup> </mrow> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", "w"]]], Rule[Editable, True]], ",", TagBox["1", Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "a"]]]], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[LerchPhi[Slot[1], Slot[2], Slot[3]]]]] </annotation> </semantics> <mo> - </mo> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> ⅇ </mi> <mi> w </mi> </msup> </mrow> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", "w"]]], Rule[Editable, True]], ",", TagBox["1", Rule[Editable, True]], ",", TagBox[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "a"]]]], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[LerchPhi[Slot[1], Slot[2], Slot[3]]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> k </ci> <ci> w </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <ci> w </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> LerchPhi </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <ci> w </ci> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> LerchPhi </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <ci> w </ci> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <leq /> <apply> <real /> <ci> w </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
