Incomplete elliptic integral of the second kind: Series representations (formula 08.04.06.0057)

EllipticE

$Mathematica Notation$

Elliptic Integrals

EllipticE[z,m]

Series representations

Generalized power series

Expansions at m==1

http://functions.wolfram.com/08.04.06.0057.01

Input Form

EllipticE[z, m] \[Proportional] (-1)^Round[Re[z]/Pi] (Sin[z] + (1/2) (Sin[z] - ArcTanh[Sin[z]]) (m - 1) + (Sec[z]^2/32) (6 ArcTanh[Sin[z]] Cos[z]^2 - 3 Sin[z] - Sin[3 z]) (m - 1)^2 + \[Ellipsis]) + 2 Round[Re[z]/Pi] (1 + Log[1 - m] ((m - 1)/4) (1 + (3 (1 - m))/8 + (15/64) (1 - m)^2 + \[Ellipsis]) + ((m - 1)/4) (1 - 4 Log[2] + ((-13 + 24 Log[2])/16) (m - 1) - ((3 (-3 + 5 Log[2]))/16) (m - 1)^2 + \[Ellipsis])) /; (m -> 1) && !Element[(Pi + 2 Re[z])/(4 Pi), Integers]

Standard Form

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MathML Form

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</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> <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> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> ¬ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <ms> π </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <ms> π </ms> </list> </apply> </apply> <ms> ∈ </ms> <apply> <ci> TagBox </ci> <ms> ℤ </ms> <apply> <ci> Function </ci> <list /> <integers /> </apply> </apply> </list> </apply> </list> </apply> </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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], "-", RowBox[List["ArcTanh", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["1", "32"], " ", SuperscriptBox[RowBox[List["Sec", "[", "z", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", RowBox[List["ArcTanh", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]], " ", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"]]], "-", RowBox[List["3", " ", RowBox[List["Sin", "[", "z", "]"]]]], "-", RowBox[List["Sin", "[", RowBox[List["3", " ", "z"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "m"]], "]"]], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]]]], "8"], "+", RowBox[List[FractionBox["15", "64"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", RowBox[List["Log", "[", "2", "]"]]]], "+", RowBox[List[FractionBox["1", "16"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "13"]], "+", RowBox[List["24", " ", RowBox[List["Log", "[", "2", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]]]], "-", RowBox[List[FractionBox["1", "16"], " ", RowBox[List["(", RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["5", " ", RowBox[List["Log", "[", "2", "]"]]]]]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["m", "\[Rule]", "1"]], ")"]], "&&", RowBox[List["!", RowBox[List[FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "\[Element]", "Integers"]]]]]]]]]]]]

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

2007-05-02

EllipticE[z]