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WeierstrassP






Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassP[z,{g2,g3}] > Representations through more general functions > Through other functions > Involving some hypergeometric-type functions





http://functions.wolfram.com/09.13.26.0001.01









  


  










Input Form





WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}] == w /; z == (-(1/Sqrt[w - Subscript[e, 1]])) AppellF1[1/2, 1/2, 1/2, 3/2, (Subscript[e, 2] - Subscript[e, 1])/(w - Subscript[e, 1]), (Subscript[e, 3] - Subscript[e, 1])/(w - Subscript[e, 1])] && 4 w^3 - Subscript[g, 2] w - Subscript[g, 3] == 4 (w - Subscript[e, 1]) (w - Subscript[e, 2]) (w - Subscript[e, 3])










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> &#8472; </mi> <mo> &#8289; </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> &#10869; </mo> <mi> w </mi> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mi> w </mi> <mo> - </mo> <msub> <mi> e </mi> <mn> 1 </mn> </msub> </mrow> </msqrt> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mrow> <msub> <mi> e </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> e </mi> <mn> 1 </mn> </msub> </mrow> <mrow> <mi> w </mi> <mo> - </mo> <msub> <mi> e </mi> <mn> 1 </mn> </msub> </mrow> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <msub> <mi> e </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> e </mi> <mn> 1 </mn> </msub> </mrow> <mrow> <mi> w </mi> <mo> - </mo> <msub> <mi> e </mi> <mn> 1 </mn> </msub> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> w </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mi> w </mi> </mrow> <mo> - </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> &#10869; </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> - </mo> <msub> <mi> e </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> - </mo> <msub> <mi> e </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> - </mo> <msub> <mi> e </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <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> <ci> w </ci> </apply> <apply> <and /> <apply> <eq /> <ci> z </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> AppellF1 </ci> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> w </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <ci> w </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> e </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










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["w", "/;", RowBox[List[RowBox[List["z", "\[Equal]", RowBox[List["-", FractionBox[RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", FractionBox[RowBox[List[SubscriptBox["e", "2"], "-", SubscriptBox["e", "1"]]], RowBox[List["w", "-", SubscriptBox["e", "1"]]]], ",", FractionBox[RowBox[List[SubscriptBox["e", "3"], "-", SubscriptBox["e", "1"]]], RowBox[List["w", "-", SubscriptBox["e", "1"]]]]]], "]"]], SqrtBox[RowBox[List["w", "-", SubscriptBox["e", "1"]]]]]]]]], "&&", RowBox[List[RowBox[List[RowBox[List["4", " ", SuperscriptBox["w", "3"]]], "-", RowBox[List[SubscriptBox["g", "2"], " ", "w"]], "-", SubscriptBox["g", "3"]]], "\[Equal]", RowBox[List["4", " ", RowBox[List["(", RowBox[List["w", "-", SubscriptBox["e", "1"]]], ")"]], " ", RowBox[List["(", RowBox[List["w", "-", SubscriptBox["e", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["w", "-", SubscriptBox["e", "3"]]], ")"]]]]]]]]]]]]]]










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





2001-10-29





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