Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











WeierstrassPPrime






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/09.14.07.0001.01









  


  










Input Form





WeierstrassPPrime[z, {Subscript[g, 2], Subscript[g, 3]}] == -(2/z^3) + (1/8) Integrate[ t^2 ((Sin[t (z/2)] E^(I t (Subscript[\[Omega], 2]/2)) Cos[t (Subscript[\[Omega], 2]/2)])/ (Sin[t ((Subscript[\[Omega], 1] - Subscript[\[Omega], 2])/2)] Sin[t ((Subscript[\[Omega], 1] + Subscript[\[Omega], 2])/2)]) + Sinh[(t z)/2] ((Cosh[t Subscript[\[Omega], 2]] + Sinh[t (Subscript[\[Omega], 2]/2)]/E^(t (Subscript[\[Omega], 2]/2)))/ (Sinh[t ((Subscript[\[Omega], 1] - Subscript[\[Omega], 2])/2)] Sinh[t ((Subscript[\[Omega], 1] + Subscript[\[Omega], 2])/2)]))), {t, 0, Infinity}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["WeierstrassPPrime", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["2", SuperscriptBox["z", "3"]]]], "+", RowBox[List[FractionBox["1", "8"], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox["t", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["t", " ", RowBox[List["z", "/", "2"]]]], "]"]], SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "t", " ", RowBox[List[SubscriptBox["\[Omega]", "2"], "/", "2"]]]]], " ", RowBox[List["Cos", "[", RowBox[List["t", " ", RowBox[List[SubscriptBox["\[Omega]", "2"], "/", "2"]]]], " ", "]"]]]], RowBox[List[RowBox[List["Sin", "[", RowBox[List["t", " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "-", SubscriptBox["\[Omega]", "2"]]], ")"]], "/", "2"]]]], " ", "]"]], " ", RowBox[List["Sin", "[", RowBox[List["t", " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "+", SubscriptBox["\[Omega]", "2"]]], ")"]], "/", "2"]]]], "]"]]]]], "+", RowBox[List[RowBox[List["Sinh", "[", FractionBox[RowBox[List["t", " ", "z"]], "2"], "]"]], FractionBox[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["t", " ", SubscriptBox["\[Omega]", "2"]]], "]"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "t"]], " ", RowBox[List[SubscriptBox["\[Omega]", "2"], "/", "2"]]]]], " ", RowBox[List["Sinh", "[", RowBox[List["t", " ", RowBox[List[SubscriptBox["\[Omega]", "2"], "/", "2"]]]], " ", "]"]], " "]]]], RowBox[List[RowBox[List["Sinh", "[", RowBox[List["t", " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "-", SubscriptBox["\[Omega]", "2"]]], ")"]], "/", "2"]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["t", " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "+", SubscriptBox["\[Omega]", "2"]]], ")"]], "/", "2"]]]], "]"]]]]]]]]], ")"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> &#8472; </mi> <mo> &#8242; </mo> </msup> <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> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 2 </mn> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mfrac> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mrow> <msup> <mi> t </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 2 </mn> </msub> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> t </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 2 </mn> </msub> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> t </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> t </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> &#969; </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> t </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> &#969; </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> t </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> t </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 2 </mn> </msub> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> t </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 2 </mn> </msub> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> t </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> &#969; </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> t </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> &#969; </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> </list> <ci> &#8472; </ci> </apply> <apply> <ci> CompoundExpression </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <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> &#969; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <ci> t </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> t </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </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> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 2 </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> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <sinh /> <apply> <times /> <ci> t </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <cosh /> <apply> <times /> <ci> t </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 2 </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'> 2 </cn> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> t </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </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; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 2 </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> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WeierstrassPPrime", "[", RowBox[List["z_", ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["2", SuperscriptBox["z", "3"]]]], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["t", " ", "z"]], "2"], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "t", " ", SubscriptBox["\[Omega]", "2"]]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["t", " ", SubscriptBox["\[Omega]", "2"]]], "2"], "]"]]]], RowBox[List[RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", "t", " ", RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "-", SubscriptBox["\[Omega]", "2"]]], ")"]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", "t", " ", RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "+", SubscriptBox["\[Omega]", "2"]]], ")"]]]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["Sinh", "[", FractionBox[RowBox[List["t", " ", "z"]], "2"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["t", " ", SubscriptBox["\[Omega]", "2"]]], "]"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["-", "t"]], ")"]], " ", SubscriptBox["\[Omega]", "2"]]]], " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["t", " ", SubscriptBox["\[Omega]", "2"]]], "2"], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["Sinh", "[", RowBox[List[FractionBox["1", "2"], " ", "t", " ", RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "-", SubscriptBox["\[Omega]", "2"]]], ")"]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List[FractionBox["1", "2"], " ", "t", " ", RowBox[List["(", RowBox[List[SubscriptBox["\[Omega]", "1"], "+", SubscriptBox["\[Omega]", "2"]]], ")"]]]], "]"]]]]]]], ")"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.