Weierstrass elliptic function: Identities (formula 09.13.17.0008)

WeierstrassP

$Mathematica Notation$

Elliptic Functions

WeierstrassP[z,{g₂,g₃}]

Identities

Functional identities

Generalization for arbitrary a, b, c, d

http://functions.wolfram.com/09.13.17.0008.01

Input Form

(4 WeierstrassP[a, {Subscript[g, 2], Subscript[g, 3]}] WeierstrassP[b, {Subscript[g, 2], Subscript[g, 3]}] WeierstrassP[c, {Subscript[g, 2], Subscript[g, 3]}] - Subscript[g, 3]) (WeierstrassP[a, {Subscript[g, 2], Subscript[g, 3]}] + WeierstrassP[b, {Subscript[g, 2], Subscript[g, 3]}] + WeierstrassP[c, {Subscript[g, 2], Subscript[g, 3]}]) - (WeierstrassP[b, {Subscript[g, 2], Subscript[g, 3]}] WeierstrassP[c, {Subscript[g, 2], Subscript[g, 3]}] + WeierstrassP[c, {Subscript[g, 2], Subscript[g, 3]}] WeierstrassP[a, {Subscript[g, 2], Subscript[g, 3]}] + WeierstrassP[a, {Subscript[g, 2], Subscript[g, 3]}] WeierstrassP[b, {Subscript[g, 2], Subscript[g, 3]}] + Subscript[g, 2]/4)^ 2 == (-(WeierstrassP[c, {Subscript[g, 2], Subscript[g, 3]}] - WeierstrassP[a, {Subscript[g, 2], Subscript[g, 3]}])^2) (WeierstrassP[b, {Subscript[g, 2], Subscript[g, 3]}] - WeierstrassP[c + a, {Subscript[g, 2], Subscript[g, 3]}]) (WeierstrassP[b, {Subscript[g, 2], Subscript[g, 3]}] - WeierstrassP[c - a, {Subscript[g, 2], Subscript[g, 3]}])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", RowBox[List["WeierstrassP", "[", RowBox[List["a", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], " ", RowBox[List["WeierstrassP", "[", RowBox[List["b", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], " ", RowBox[List["WeierstrassP", "[", RowBox[List["c", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "-", SubscriptBox["g", "3"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["WeierstrassP", "[", RowBox[List["a", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "+", RowBox[List["WeierstrassP", "[", RowBox[List["b", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "+", RowBox[List["WeierstrassP", "[", RowBox[List["c", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], ")"]]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["WeierstrassP", "[", RowBox[List["b", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], " ", RowBox[List["WeierstrassP", "[", RowBox[List["c", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "+", RowBox[List[RowBox[List["WeierstrassP", "[", RowBox[List["c", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], " ", RowBox[List["WeierstrassP", "[", RowBox[List["a", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "+", RowBox[List[RowBox[List["WeierstrassP", "[", RowBox[List["a", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], " ", RowBox[List["WeierstrassP", "[", RowBox[List["b", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "+", FractionBox[SubscriptBox["g", "2"], "4"]]], ")"]], "2"]]], "\[Equal]", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["WeierstrassP", "[", RowBox[List["c", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "-", RowBox[List["WeierstrassP", "[", RowBox[List["a", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], ")"]], "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["WeierstrassP", "[", RowBox[List["b", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "-", RowBox[List["WeierstrassP", "[", RowBox[List[RowBox[List["c", "+", "a"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["WeierstrassP", "[", RowBox[List["b", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "-", RowBox[List["WeierstrassP", "[", RowBox[List[RowBox[List["c", "-", "a"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], ")"]]]]]]]]

MathML Form

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</msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> WeierstrassP </ci> <ci> a </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> <ci> WeierstrassP </ci> <ci> b </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> <ci> WeierstrassP </ci> <ci> c </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> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> WeierstrassP </ci> <ci> a </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> <ci> WeierstrassP </ci> <ci> b </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> <ci> WeierstrassP </ci> <ci> c </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> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <ci> WeierstrassP </ci> <ci> a </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> <ci> WeierstrassP </ci> <ci> b </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> <apply> <times /> <apply> <ci> WeierstrassP </ci> <ci> a </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> <ci> WeierstrassP </ci> <ci> c </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> <apply> <times /> <apply> <ci> WeierstrassP </ci> <ci> b </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> <ci> WeierstrassP </ci> <ci> c </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> <apply> <times /> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <ci> WeierstrassP </ci> <ci> c </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 /> <cn type='integer'> -1 </cn> <apply> <ci> WeierstrassP </ci> <ci> a </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> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> WeierstrassP </ci> <ci> b </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 /> <cn type='integer'> -1 </cn> <apply> <ci> WeierstrassP </ci> <apply> <plus /> <ci> a </ci> <ci> c </ci> </apply> <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> </apply> <apply> <plus /> <apply> <ci> WeierstrassP </ci> <ci> b </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 /> <cn type='integer'> -1 </cn> <apply> <ci> WeierstrassP </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <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> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29