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http://functions.wolfram.com/09.15.21.0001.01
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Integrate[WeierstrassSigma[z, {Subscript[g, 2], Subscript[g, 3]}], z] ==
(Subscript[\[Omega], 1]^(3/2)/Sqrt[2 Pi Subscript[\[Eta], 1]])
Product[1/(1 - q^(2 m)), {m, 1, Infinity}]^3
Sum[(-1)^k q^(k (k + 1)) Exp[(Pi^2 (2 k + 1)^2)/(8 Subscript[\[Eta], 1]
Subscript[\[Omega], 1])]
(Erf[(Pi (2 k + 1) + 2 I z Subscript[\[Eta], 1])/
(2 Sqrt[2 Subscript[\[Eta], 1] Subscript[\[Omega], 1]])] +
Erf[(Pi (2 k + 1) - 2 I z Subscript[\[Eta], 1])/
(2 Sqrt[2 Subscript[\[Eta], 1] Subscript[\[Omega], 1]])]),
{k, 0, Infinity}]
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["WeierstrassSigma", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[SubsuperscriptBox["\[Omega]", "1", RowBox[List["3", "/", "2"]]], SqrtBox[RowBox[List["2", " ", "\[Pi]", " ", SubscriptBox["\[Eta]", "1"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["m", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", "m"]]]]]]]], ")"]], "3"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["q", RowBox[List["k", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]]]]], " ", RowBox[List["Exp", "[", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], "2"]]], RowBox[List["8", " ", SubscriptBox["\[Eta]", "1"], " ", SubscriptBox["\[Omega]", "1"]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["\[Pi]", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", SubscriptBox["\[Eta]", "1"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", SubscriptBox["\[Eta]", "1"], SubscriptBox["\[Omega]", "1"]]]]]]], "]"]], "+", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["\[Pi]", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]], "-", RowBox[List["2", "\[ImaginaryI]", " ", "z", " ", SubscriptBox["\[Eta]", "1"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", SubscriptBox["\[Eta]", "1"], SubscriptBox["\[Omega]", "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> <mo> ∫ </mo> <mrow> <semantics> <mrow> <mi> σ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox["z", Rule[Editable, True]], ";", TagBox[SubscriptBox["g", "2"], Rule[Editable, True]]]], ",", TagBox[SubscriptBox["g", "3"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <msubsup> <mi> ω </mi> <mn> 1 </mn> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msubsup> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> η </mi> <mn> 1 </mn> </msub> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> q </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <msup> <mi> q </mi> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msub> <mi> η </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> erf </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <msub> <mi> η </mi> <mn> 1 </mn> </msub> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> η </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> erf </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <msub> <mi> η </mi> <mn> 1 </mn> </msub> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> η </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <ci> WeierstrassSigma </ci> <ci> z </ci> <list> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <apply> <ci> Subscript </ci> <ci> η </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <product /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <ci> q </ci> <apply> <times /> <ci> k </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <exp /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <ci> Subscript </ci> <ci> η </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> Erf </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> η </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> η </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Erf </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> η </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> η </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["WeierstrassSigma", "[", RowBox[List["z_", ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SubsuperscriptBox["\[Omega]", "1", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["m", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", " ", "m"]]]]]]]], ")"]], "3"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["q", RowBox[List["k", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "2"]]], RowBox[List["8", " ", SubscriptBox["\[Eta]", "1"], " ", SubscriptBox["\[Omega]", "1"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", SubscriptBox["\[Eta]", "1"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", SubscriptBox["\[Eta]", "1"], " ", SubscriptBox["\[Omega]", "1"]]]]]]], "]"]], "+", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z", " ", SubscriptBox["\[Eta]", "1"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", SubscriptBox["\[Eta]", "1"], " ", SubscriptBox["\[Omega]", "1"]]]]]]], "]"]]]], ")"]]]]]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]", " ", SubscriptBox["\[Eta]", "1"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
