Incomplete elliptic integral of the second kind: Series representations (formula 08.04.06.0069)

EllipticE

$Mathematica Notation$

Elliptic Integrals

EllipticE[z,m]

Series representations

Generalized power series

Expansions at m==infinity NEW E

http://functions.wolfram.com/08.04.06.0069.01

Input Form

EllipticE[z, m] \[Proportional] 4 (Sqrt[-m] + (1 + Log[-16 m])/(4 Sqrt[-m])) Round[Re[z]/(2 Pi)] + Sqrt[(-m) Sin[z]^2] Tan[z/2] + ((-(Csc[z]^2/(2 m)) + Log[-4 m Sin[z]^2]) Sin[z])/ (4 Sqrt[(-m) Sin[z]^2]) - ((Sin[z] Sqrt[(-m) Sin[z]^2])/(16 m)) HypergeometricPFQ[{1, 1, 3/2}, {2, 3}, Sin[z]^2] /; (Abs[m] -> Infinity) && Re[z] != Pi (2 k + 1) && Element[k, Integers]

Standard Form

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MathML Form

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</ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 32 </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sec </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 6 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> tanh </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> cos </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <ms> … </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> ⌊ </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> π </ms> </apply> <ms> ⌉ </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 4 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - 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</ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <ms> 2 </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 16 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 24 </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <ms> 2 </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <ms> 13 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 16 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 5 </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <ms> 2 </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <ms> 3 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <ms> … </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms>  </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["4", " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["-", "m"]]], "+", FractionBox[RowBox[List["1", "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "16"]], " ", "m"]], "]"]]]], RowBox[List["4", " ", SqrtBox[RowBox[List["-", "m"]]]]]]]], ")"]], " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], " ", RowBox[List["Tan", "[", FractionBox["z", "2"], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], RowBox[List["2", " ", "m"]]]]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "4"]], " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]], ")"]], " ", RowBox[List["Sin", "[", "z", "]"]]]], RowBox[List["4", " ", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", "3"]], "}"]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]], RowBox[List["16", " ", "m"]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[NotEqual]", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]]]]]]]]]]

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

2007-05-02

EllipticE[z]