Weierstrass invariants: Differentiation (formula 09.19.20.0010)

WeierstrassInvariants

$Mathematica Notation$

Elliptic Functions

WeierstrassInvariants[{w₁,w₃}]

Differentiation

Fractional integro-differentiation

With respect to omega₃

http://functions.wolfram.com/09.19.20.0010.01

Input Form

D[WeierstrassInvariants[{Subscript[\[Omega], 1], Subscript[\[Omega], 3]}], {Subscript[\[Omega], 3], \[Alpha]}] == {(Pi^4/12) FDPowerConstant[Subscript[\[Omega], 3], -4, \[Alpha]] Subscript[\[Omega], 3]^(-4 - \[Alpha]) + (15/(Subscript[\[Omega], 3]^\[Alpha] (2 Subscript[\[Omega], 1]^4))) Sum[(1/m^4) Hypergeometric2F1Regularized[1, 4, 1 - \[Alpha], -((n Subscript[\[Omega], 3])/(m Subscript[\[Omega], 1]))], {n, -Infinity, Infinity}, {m, 1, Infinity}], (Pi^6/216) FDPowerConstant[Subscript[\[Omega], 3], -6, \[Alpha]] Subscript[\[Omega], 3]^(-6 - \[Alpha]) + (35/(Subscript[\[Omega], 3]^\[Alpha] (8 Subscript[\[Omega], 1]^6))) Sum[(1/m^6) Hypergeometric2F1Regularized[1, 6, 1 - \[Alpha], -((n Subscript[\[Omega], 3])/(m Subscript[\[Omega], 1]))], {n, -Infinity, Infinity}, {m, 1, Infinity}]}

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "3"], ",", "\[Alpha]"]], "}"]]], RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]]]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "4"], "12"], RowBox[List["FDPowerConstant", "[", RowBox[List[SubscriptBox["\[Omega]", "3"], ",", RowBox[List["-", "4"]], ",", "\[Alpha]"]], "]"]], SubsuperscriptBox["\[Omega]", "3", RowBox[List[RowBox[List["-", "4"]], "-", "\[Alpha]"]]]]], "+", RowBox[List[FractionBox[RowBox[List["15", " ", SubsuperscriptBox["\[Omega]", "3", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List["2", " ", SubsuperscriptBox["\[Omega]", "1", "4"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox["1", SuperscriptBox["m", "4"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "4", ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["-", FractionBox[RowBox[List["n", " ", SubscriptBox["\[Omega]", "3"]]], RowBox[List["m", " ", SubscriptBox["\[Omega]", "1"]]]]]]]], "]"]]]]]]]]]]]], ",", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "6"], "216"], RowBox[List["FDPowerConstant", "[", RowBox[List[SubscriptBox["\[Omega]", "3"], ",", RowBox[List["-", "6"]], ",", "\[Alpha]"]], "]"]], SubsuperscriptBox["\[Omega]", "3", RowBox[List[RowBox[List["-", "6"]], "-", "\[Alpha]"]]]]], "+", RowBox[List[FractionBox[RowBox[List["35", " ", SubsuperscriptBox["\[Omega]", "3", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List["8", SubsuperscriptBox["\[Omega]", "1", "6"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox["1", SuperscriptBox["m", "6"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "6", ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["-", FractionBox[RowBox[List["n", " ", SubscriptBox["\[Omega]", "3"]]], RowBox[List["m", " ", SubscriptBox["\[Omega]", "1"]]]]]]]], "]"]]]]]]]]]]]]]], "}"]]]]]]

MathML Form

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<ci> Subscript </ci> <ci> ℱ𝒞 </ci> <ci> exp </ci> </apply> <ci> α </ci> </apply> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -4 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <cn type='integer'> -4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <cn type='integer'> 1 </cn> <cn type='integer'> 4 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> ω </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> 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Rule Form

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

2001-10-29