Weierstrass invariants: Differentiation (formula 09.19.20.0002)

WeierstrassInvariants

$Mathematica Notation$

Elliptic Functions

WeierstrassInvariants[{w₁,w₃}]

Differentiation

Low-order differentiation

With respect to omega₁

http://functions.wolfram.com/09.19.20.0002.01

Input Form

D[WeierstrassInvariants[{Subscript[\[Omega], 1], Subscript[\[Omega], 3]}], Subscript[\[Omega], 1]] == {-((4 Subscript[g, 2])/Subscript[\[Omega], 1]) - ((40 I Pi^5 Subscript[\[Omega], 3])/Subscript[\[Omega], 1]^6) Sum[(k^4 q^(2 k))/(1 - q^(2 k))^2, {k, 1, Infinity}], -((6 Subscript[g, 3])/Subscript[\[Omega], 1]) + ((14 I Pi^7 Subscript[\[Omega], 3])/(3 Subscript[\[Omega], 1]^8)) Sum[(k^6 q^(2 k))/(1 - q^(2 k))^2, {k, 1, Infinity}]} /; q == Exp[Pi I (Subscript[\[Omega], 3]/Subscript[\[Omega], 1])]

Standard Form

Cell[BoxData[RowBox[List[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[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["4", " ", SubscriptBox["g", "2"]]], SubscriptBox["\[Omega]", "1"]]]], "-", RowBox[List[FractionBox[RowBox[List["40", "\[ImaginaryI]", " ", SuperscriptBox["\[Pi]", "5"], " ", SubscriptBox["\[Omega]", "3"]]], RowBox[List[" ", SubsuperscriptBox["\[Omega]", "1", "6"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["k", "4"], " ", SuperscriptBox["q", RowBox[List["2", " ", "k"]]]]], RowBox[List[" ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", " ", "k"]]]]], ")"]], "2"]]]]]]]]]], ",", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["6", " ", SubscriptBox["g", "3"]]], SubscriptBox["\[Omega]", "1"]]]], "+", RowBox[List[FractionBox[RowBox[List["14", "\[ImaginaryI]", " ", SuperscriptBox["\[Pi]", "7"], SubscriptBox["\[Omega]", "3"]]], RowBox[List["3", " ", SubsuperscriptBox["\[Omega]", "1", "8"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["k", "6"], SuperscriptBox["q", RowBox[List["2", " ", "k"]]], " "]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", " ", "k"]]]]], ")"]], "2"]]]]]]]]]], "}"]]]], "/;", RowBox[List["q", "\[Equal]", RowBox[List["Exp", "[", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", FractionBox[SubscriptBox["\[Omega]", "3"], SubscriptBox["\[Omega]", "1"]]]], "]"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <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> <mo> { </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 40 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 5 </mn> </msup> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <msubsup> <mi> ω </mi> <mn> 1 </mn> <mn> 6 </mn> </msubsup> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msup> <mi> k </mi> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> q </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> q </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 14 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 7 </mn> </msup> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> </mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msubsup> <mi> ω </mi> <mn> 1 </mn> <mn> 8 </mn> </msubsup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msup> <mi> k </mi> <mn> 6 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> q </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> q </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> q </mi> <mo> ⩵ </mo> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mfrac> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <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> <list> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 40 </cn> <imaginaryi /> <apply> <power /> <pi /> <cn type='integer'> 5 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> k </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 14 </cn> <imaginaryi /> <apply> <power /> <pi /> <cn type='integer'> 7 </cn> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 8 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> k </ci> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </list> </apply> <apply> <eq /> <ci> q </ci> <apply> <exp /> <apply> <times /> <pi /> <imaginaryi /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </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[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["4", " ", SubscriptBox["g", "2"]]], SubscriptBox["\[Omega]\[Omega]", "1"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["40", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[Pi]", "5"], " ", SubscriptBox["\[Omega]\[Omega]", "3"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["k", "4"], " ", SuperscriptBox["q", RowBox[List["2", " ", "k"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", " ", "k"]]]]], ")"]], "2"]]]]]], SubsuperscriptBox["\[Omega]\[Omega]", "1", "6"]]]], ",", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["6", " ", SubscriptBox["g", "3"]]], SubscriptBox["\[Omega]\[Omega]", "1"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["14", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[Pi]", "7"], " ", SubscriptBox["\[Omega]\[Omega]", "3"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["k", "6"], " ", SuperscriptBox["q", RowBox[List["2", " ", "k"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["q", RowBox[List["2", " ", "k"]]]]], ")"]], "2"]]]]]], RowBox[List["3", " ", SubsuperscriptBox["\[Omega]\[Omega]", "1", "8"]]]]]]]], "}"]], "/;", RowBox[List["q", "\[Equal]", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", SubscriptBox["\[Omega]\[Omega]", "3"]]], SubscriptBox["\[Omega]\[Omega]", "1"]]]]]]]]]]]

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

2001-10-29