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WeierstrassP






Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassP[z,{g2,g3}] > Representations through equivalent functions > With related functions > Involving other related functions





http://functions.wolfram.com/09.13.27.0013.01









  


  










Input Form





WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}] == (x + a/3)/4^3^(-1) /; {x, y} == EllipticExp[z/2^3^(-1), {a, b}] && a == (I 3^(1/6) (I (3 I + Sqrt[3]) Subscript[g, 2] + 3^(1/6) (I + Sqrt[3]) (9 Subscript[g, 3] + Sqrt[-3 Subscript[g, 2]^3 + 81 Subscript[g, 3]^2])^(2/3)))/ (2 2^(1/3) (9 Subscript[g, 3] + Sqrt[-3 Subscript[g, 2]^3 + 81 Subscript[g, 3]^2])^(1/3)) && b == (I 3^(1/3) (I + Sqrt[3]) Subscript[g, 2]^2 - 3^(1/6) (3 (3 I + Sqrt[3]) Subscript[g, 3] + Sqrt[-Subscript[g, 2]^3 + 27 Subscript[g, 3]^2] + I Sqrt[-3 Subscript[g, 2]^3 + 81 Subscript[g, 3]^2]) (9 Subscript[g, 3] + Sqrt[-3 Subscript[g, 2]^3 + 81 Subscript[g, 3]^2])^ (1/3) + 2 Subscript[g, 2] (9 Subscript[g, 3] + Sqrt[-3 Subscript[g, 2]^3 + 81 Subscript[g, 3]^2])^(2/3))/ (2 2^(2/3) (9 Subscript[g, 3] + Sqrt[-3 Subscript[g, 2]^3 + 81 Subscript[g, 3]^2])^(2/3))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WeierstrassP", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["4", RowBox[List[RowBox[List["-", "1"]], "/", "3"]]], RowBox[List["(", RowBox[List["x", "+", FractionBox["a", "3"]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["x", ",", "y"]], "}"]], "\[Equal]", RowBox[List["EllipticExp", "[", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "/", "3"]]], "z"]], ",", RowBox[List["{", RowBox[List["a", ",", "b"]], "}"]]]], "]"]]]], "\[And]", RowBox[List["a", "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]], " ", SubscriptBox["g", "2"]]], "+", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["9", " ", SubscriptBox["g", "3"]]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SubsuperscriptBox["g", "2", "3"]]], "+", RowBox[List["81", " ", SubsuperscriptBox["g", "3", "2"]]]]]]]], ")"]], RowBox[List["2", "/", "3"]]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["9", " ", SubscriptBox["g", "3"]]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SubsuperscriptBox["g", "2", "3"]]], "+", RowBox[List["81", " ", SubsuperscriptBox["g", "3", "2"]]]]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], ")"]]]]]], "\[And]", RowBox[List["b", "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SubsuperscriptBox["g", "2", "2"]]], "-", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]], " ", SubscriptBox["g", "3"]]], "+", SqrtBox[RowBox[List[RowBox[List["-", " ", SubsuperscriptBox["g", "2", "3"]]], "+", RowBox[List["27", " ", SubsuperscriptBox["g", "3", "2"]]]]]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SubsuperscriptBox["g", "2", "3"]]], "+", RowBox[List["81", " ", SubsuperscriptBox["g", "3", "2"]]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["9", " ", SubscriptBox["g", "3"]]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SubsuperscriptBox["g", "2", "3"]]], "+", RowBox[List["81", " ", SubsuperscriptBox["g", "3", "2"]]]]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", RowBox[List["2", " ", SubscriptBox["g", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["9", " ", SubscriptBox["g", "3"]]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SubsuperscriptBox["g", "2", "3"]]], "+", RowBox[List["81", " ", SubsuperscriptBox["g", "3", "2"]]]]]]]], ")"]], RowBox[List["2", "/", "3"]]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["9", " ", SubscriptBox["g", "3"]]], "+", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SubsuperscriptBox["g", "2", "3"]]], "+", RowBox[List["81", " ", SubsuperscriptBox["g", "3", "2"]]]]]]]], ")"]], RowBox[List["2", "/", "3"]]]]], ")"]]]]]]]]]]]]










MathML Form







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<mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mroot> <mn> 3 </mn> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> + </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msubsup> <mi> g </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> + </mo> <msqrt> <mrow> <mrow> <mn> 81 </mn> <mo> &#8290; </mo> <msubsup> <mi> g </mi> <mn> 3 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msubsup> <mi> g </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </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> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> x </ci> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <list> <ci> x </ci> <ci> y </ci> </list> <apply> <ci> eexp </ci> <apply> <ci> CompoundExpression </ci> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> <ci> a </ci> </apply> <ci> b </ci> </apply> </apply> <apply> <eq /> <ci> a </ci> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 6 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 6 </cn> </apply> <apply> <plus /> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 81 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <imaginaryi /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 81 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <ci> b </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 6 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 27 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 81 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 81 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 81 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <plus /> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <imaginaryi /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 81 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





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