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http://functions.wolfram.com/10.07.07.0004.01
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PolyLog[2, z] == Integrate[(E^(t/z)/t) Sum[(-t)^k/(k k!), {k, 1, Infinity}],
{t, 0, Infinity}] /; Re[z] < 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["t", "/", "z"]]], "t"], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "t"]], ")"]], "k"], RowBox[List["k", " ", RowBox[List["k", "!"]]]]]]], ")"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "<", "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> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mfrac> <msup> <mi> ⅇ </mi> <mrow> <mi> t </mi> <mo> / </mo> <mi> z </mi> </mrow> </msup> <mi> t </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> t </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["t", "/", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "t"]], ")"]], "k"], RowBox[List["k", " ", RowBox[List["k", "!"]]]]]]]]], "t"], RowBox[List["\[DifferentialD]", "t"]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "<", "0"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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