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WeierstrassP






Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassP[z,{g2,g3}] > Integral representations > On the real axis > Of the direct function





http://functions.wolfram.com/09.13.07.0001.01









  


  










Input Form





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}]










Standard Form





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MathML Form







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</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> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> t </ci> <apply> <ci> Subscript </ci> <ci> &#969; </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> &#969; 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Rule Form





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"]]]]]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29