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http://functions.wolfram.com/10.06.04.0043.01
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Limit[LerchPhi[x + I \[Epsilon], s, a], \[Epsilon] -> Plus[0]] ==
LerchPhi[x, s, a] + ((2 I Pi)/(x^a Gamma[s])) Log[x]^(s - 1) /; x > 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["LerchPhi", "[", RowBox[List[RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]"]]]], ",", "s", ",", "a"]], "]"]], ",", RowBox[List["\[Epsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "\[Equal]", " ", RowBox[List[RowBox[List["LerchPhi", "[", RowBox[List["x", ",", "s", ",", "a"]], "]"]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", SuperscriptBox["x", RowBox[List["-", "a"]]]]], " "]], RowBox[List["Gamma", "[", "s", "]"]]], SuperscriptBox[RowBox[List["Log", "[", "x", "]"]], RowBox[List["s", "-", "1"]]]]]]]]], "/;", RowBox[List["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> ϵ </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ϵ </mi> </mrow> </mrow> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox[RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]"]]]], Rule[Editable, True]], ",", TagBox["s", Rule[Editable, True]], ",", TagBox["a", Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2, $CellContext`e3], LerchPhi[$CellContext`e1, $CellContext`e2, $CellContext`e3]]]] </annotation> </semantics> </mrow> <mo> ⩵ </mo> <mrow> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </mo> <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["\[CapitalPhi]", "(", RowBox[List[TagBox["x", Rule[Editable, True]], ",", TagBox["s", Rule[Editable, True]], ",", TagBox["a", Rule[Editable, True]]]], ")"]], 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> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> x </mi> <mrow> <mo> - </mo> <mi> a </mi> </mrow> </msup> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </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> > </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> ϵ </ci> </bvar> <condition> <apply> <tendsto /> <ci> ϵ </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <ci> LerchPhi </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> ϵ </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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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[RowBox[List["LerchPhi", "[", RowBox[List[RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]_"]]]], ",", "s_", ",", "a_"]], "]"]], ",", RowBox[List["\[Epsilon]_", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["LerchPhi", "[", RowBox[List["x", ",", "s", ",", "a"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", SuperscriptBox["x", RowBox[List["-", "a"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Log", "[", "x", "]"]], RowBox[List["s", "-", "1"]]]]], RowBox[List["Gamma", "[", "s", "]"]]]]], "/;", RowBox[List["x", ">", "1"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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