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http://functions.wolfram.com/09.13.07.0001.01
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WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}] ==
1/z^2 + (1/4) Integrate[
t ((E^(I t (Subscript[\[Omega], 3]/2)) (1 - Cos[t (z/2)])
Cos[t (Subscript[\[Omega], 3]/2)])/
(Sin[t ((Subscript[\[Omega], 1] - Subscript[\[Omega], 3])/2)]
Sin[t ((Subscript[\[Omega], 1] + Subscript[\[Omega], 3])/2)]) +
(Cosh[(t z)/2] - 1) ((Cosh[t Subscript[\[Omega], 3]] +
Sinh[t (Subscript[\[Omega], 3]/2)]/E^(t (Subscript[\[Omega], 3]/2)))/
(Sinh[t ((Subscript[\[Omega], 1] - Subscript[\[Omega], 3])/2)]
Sinh[t ((Subscript[\[Omega], 1] + Subscript[\[Omega], 3])/2)]))),
{t, 0, Infinity}]
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Cell[BoxData[RowBox[List[RowBox[List["WeierstrassP", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", SuperscriptBox["z", "2"]], "+", RowBox[List[FractionBox["1", "4"], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List["t", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "t", " ", RowBox[List[SubscriptBox["\[Omega]", "3"], "/", "2"]]]]], RowBox[List["(", RowBox[List["1", "-", RowBox[List["Cos", "[", RowBox[List["t", " ", RowBox[List["z", "/", "2"]]]], "]"]]]], ")"]], RowBox[List["Cos", "[", RowBox[List["t", " ", RowBox[List[SubscriptBox["\[Omega]", "3"], "/", "2"]]]], " ", "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List["t", " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "-", SubscriptBox["\[Omega]", "3"]]], ")"]], "/", "2"]]]], " ", "]"]], " ", RowBox[List["Sin", "[", RowBox[List["t", " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "+", SubscriptBox["\[Omega]", "3"]]], ")"]], "/", "2"]]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["t", " ", "z"]], "2"], "]"]], "-", "1"]], ")"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["t", " ", SubscriptBox["\[Omega]", "3"]]], "]"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "t"]], " ", RowBox[List[SubscriptBox["\[Omega]", "3"], "/", "2"]]]]], " ", RowBox[List["Sinh", "[", RowBox[List["t", " ", RowBox[List[SubscriptBox["\[Omega]", "3"], "/", "2"]]]], " ", "]"]]]]]], " ", ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Sinh", "[", RowBox[List["t", " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "-", SubscriptBox["\[Omega]", "3"]]], ")"]], "/", "2"]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["t", " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "+", SubscriptBox["\[Omega]", "3"]]], ")"]], "/", "2"]]]], "]"]]]], ")"]]]]]]]], ")"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]]]]]]
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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> <mi> ℘ </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> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <mi> t </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> t </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> t </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> t </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> t </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> t </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> t </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> t </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> t </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> t </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> t </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> WeierstrassP </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> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <ci> t </ci> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <cosh /> <apply> <times /> <ci> t </ci> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> t </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> t </ci> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <cosh /> <apply> <times /> <ci> t </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <sinh /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> t </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> t </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <imaginaryi /> <ci> t </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <times /> <ci> t </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <ci> t </ci> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <sin /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> t </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> t </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </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["WeierstrassP", "[", RowBox[List["z_", ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", SuperscriptBox["z", "2"]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List["t", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "t", " ", SubscriptBox["\[Omega]", "3"]]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Cos", "[", FractionBox[RowBox[List["t", " ", "z"]], "2"], "]"]]]], ")"]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["t", " ", SubscriptBox["\[Omega]", "3"]]], "2"], "]"]]]], RowBox[List[RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", "t", " ", RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "-", SubscriptBox["\[Omega]", "3"]]], ")"]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", "t", " ", RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "+", SubscriptBox["\[Omega]", "3"]]], ")"]]]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["t", " ", "z"]], "2"], "]"]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["t", " ", SubscriptBox["\[Omega]", "3"]]], "]"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["-", "t"]], ")"]], " ", SubscriptBox["\[Omega]", "3"]]]], " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["t", " ", SubscriptBox["\[Omega]", "3"]]], "2"], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["Sinh", "[", RowBox[List[FractionBox["1", "2"], " ", "t", " ", RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "-", SubscriptBox["\[Omega]", "3"]]], ")"]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List[FractionBox["1", "2"], " ", "t", " ", RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "+", SubscriptBox["\[Omega]", "3"]]], ")"]]]], "]"]]]]]]], ")"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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