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http://functions.wolfram.com/10.09.06.0004.01
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PolyLog[\[Nu], 1, z] == Gamma[-\[Nu]]
Sum[(2 Pi I k - Log[z])^\[Nu] E^(2 Pi k I), {k, -Infinity, Infinity}] /;
Re \[Nu] < -1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["\[Nu]", ",", "1", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "k"]], "-", RowBox[List["Log", "[", "z", "]"]]]], ")"]], "\[Nu]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Pi]", " ", "k", " ", "\[ImaginaryI]"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Re", " ", "\[Nu]"]], "<", RowBox[List["-", "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> <msub> <semantics> <mi> S </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mrow> <mi> ν </mi> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mi> Γ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> </mrow> <mi> ∞ </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> < </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> PolyLog </ci> <ci> ν </ci> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <apply> <times /> <ci> Γ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> z </ci> </apply> </apply> </apply> <ci> ν </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> k </ci> <imaginaryi /> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <times /> <real /> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["\[Nu]_", ",", "1", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "k"]], "-", RowBox[List["Log", "[", "z", "]"]]]], ")"]], "\[Nu]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Pi]", " ", "k", " ", "\[ImaginaryI]"]]]]]]]]], "/;", RowBox[List[RowBox[List["Re", " ", "\[Nu]"]], "<", RowBox[List["-", "1"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
