Weierstrass invariants: Differentiation (formula 09.19.20.0015)

WeierstrassInvariants

$Mathematica Notation$

Elliptic Functions

WeierstrassInvariants[{w₁,w₃}]

Differentiation

Low-order differentiation

With respect to omega₁

http://functions.wolfram.com/09.19.20.0015.01

Input Form

D[WeierstrassInvariants[{Subscript[\[Omega], 1], Subscript[\[Omega], 3]}], Subscript[\[Omega], 1]] == (-(Subscript[\[Omega], 1]/(Pi Subscript[\[Omega], 3]))) Sqrt[-(Subscript[\[Omega], 3]^2/Subscript[\[Omega], 1]^2)] ({12, 2/3} Reverse[WeierstrassInvariants[{Subscript[\[Omega], 1], Subscript[\[Omega], 3]}]^{2, 1}] Subscript[\[Omega], 3] - {8, 12} WeierstrassInvariants[{Subscript[\[Omega], 1], Subscript[\[Omega], 3]}] WeierstrassZeta[Subscript[\[Omega], 3], WeierstrassInvariants[{Subscript[\[Omega], 1], Subscript[\[Omega], 3]}]])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", SubscriptBox["\[Omega]", "1"]], RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["\[Omega]", "1"], RowBox[List["\[Pi]", " ", SubscriptBox["\[Omega]", "3"]]]]]], " ", SqrtBox[RowBox[List["-", FractionBox[SubsuperscriptBox["\[Omega]", "3", "2"], SubsuperscriptBox["\[Omega]", "1", "2"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["12", ",", FractionBox["2", "3"]]], "}"]], " ", RowBox[List["Reverse", "[", SuperscriptBox[RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]], RowBox[List["{", RowBox[List["2", ",", "1"]], "}"]]], "]"]], " ", SubscriptBox["\[Omega]", "3"]]], "-", RowBox[List[RowBox[List["{", RowBox[List["8", ",", "12"]], "}"]], " ", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]], " ", RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["\[Omega]", "3"], ",", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]]]], "]"]]]]]], ")"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <mo> ∂ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> </mfrac> <mo>  </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <msubsup> <mi> ω </mi> <mn> 3 </mn> <mn> 2 </mn> </msubsup> <msubsup> <mi> ω </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mn> 12 </mn> <mo> , </mo> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> } </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Reverse </mi> <mo> [ </mo> <msup> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> } </mo> </mrow> <mrow> <mo> { </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> </msup> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> - </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mn> 8 </mn> <mo> , </mo> <mn> 12 </mn> </mrow> <mo> } </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> } </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <semantics> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["\[Omega]", "3"], Rule[Editable, True]] </annotation> </semantics> <mo> ; </mo> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> D </ci> <list> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </list> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <list> <cn type='integer'> 12 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </list> <apply> <ci> Reverse </ci> <apply> <power /> <list> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </list> <list> <cn type='integer'> 2 </cn> <cn type='integer'> 1 </cn> </list> </apply> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <list> <cn type='integer'> 8 </cn> <cn type='integer'> 12 </cn> </list> <list> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </list> <apply> <ci> Zeta </ci> <apply> <ci> CompoundExpression </ci> <apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[SubscriptBox["\[Omega]_", "1"]]]], RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]_", "1"], ",", SubscriptBox["\[Omega]_", "3"]]], "}"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SubscriptBox["\[Omega]\[Omega]", "1"], " ", SqrtBox[RowBox[List["-", FractionBox[SubsuperscriptBox["\[Omega]\[Omega]", "3", "2"], SubsuperscriptBox["\[Omega]\[Omega]", "1", "2"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["12", ",", FractionBox["2", "3"]]], "}"]], " ", RowBox[List["Reverse", "[", SuperscriptBox[RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]\[Omega]", "1"], ",", SubscriptBox["\[Omega]\[Omega]", "3"]]], "}"]], "]"]], RowBox[List["{", RowBox[List["2", ",", "1"]], "}"]]], "]"]], " ", SubscriptBox["\[Omega]\[Omega]", "3"]]], "-", RowBox[List[RowBox[List["{", RowBox[List["8", ",", "12"]], "}"]], " ", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]\[Omega]", "1"], ",", SubscriptBox["\[Omega]\[Omega]", "3"]]], "}"]], "]"]], " ", RowBox[List["WeierstrassZeta", "[", RowBox[List[SubscriptBox["\[Omega]\[Omega]", "3"], ",", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]\[Omega]", "1"], ",", SubscriptBox["\[Omega]\[Omega]", "3"]]], "}"]], "]"]]]], "]"]]]]]], ")"]]]], RowBox[List["\[Pi]", " ", SubscriptBox["\[Omega]\[Omega]", "3"]]]]]]]]]]

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

2007-05-02