Incomplete elliptic integral of the second kind: Series representations (formula 08.04.06.0034)

EllipticE

$Mathematica Notation$

Elliptic Integrals

EllipticE[z,m]

Series representations

Generalized power series

Expansions at z==-csc^-1(m^1/2)+Pi u/;u ∈ Z

http://functions.wolfram.com/08.04.06.0034.01

Input Form

EllipticE[z, m] == (-Sqrt[m]) (EllipticE[1/m] + (-1 + 1/m) EllipticK[1/m]) + 2 u EllipticE[m] + 2 Sqrt[2] Sqrt[Sqrt[-1 + m] (z - Subscript[z, 0])] (z - Subscript[z, 0]) Sum[(((-1)^k Binomial[k - 1/2, k])/(3 + 2 k)) Sum[(((-1)^j Binomial[k, j])/(1 - 2 j)) Subscript[p, j, k] (z - Subscript[z, 0])^k, {j, 0, k}], {k, 0, Infinity}] /; Subscript[z, 0] == -ArcCsc[Sqrt[m]] + Pi u && Element[u, Integers] && Subscript[a, 0] == 1 && Subscript[a, 2 k] == ((-1)^k 2^(2 k))/(2 k + 1)! && Subscript[a, 2 k + 1] == ((-1)^(k - 1) 2^(2 k) (2 - m))/ (Sqrt[m - 1] (2 k + 2)!) && Element[k, Integers] && k >= 0 && Subscript[p, u, 0] == 1 && Subscript[p, u, v] == (1/v) Sum[(u j - v + j) Subscript[a, j] Subscript[p, u, v - j], {j, 1, v}]

Standard Form

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MathML Form

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Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo>  </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> a </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msub> <mo>  </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo>  </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> 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<mrow> <mo> ( </mo> <mrow> <mrow> <mi> u </mi> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> j </mi> <mo> - </mo> <mi> v </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> ⁢ </mo> <msub> <mi> p </mi> <mrow> <mi> u </mi> <mo> , </mo> <mrow> <mi> v </mi> <mo> - </mo> <mi> j </mi> </mrow> </mrow> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EllipticE </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> u </ci> <apply> <ci> EllipticE </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <ci> EllipticE </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> 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<ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> k </ci> <ci> ℕ </ci> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> u </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> u </ci> <ci> v </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> v </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> v </ci> </uplimit> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> u </ci> <ci> j </ci> </apply> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> v </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> u </ci> <apply> <plus /> <ci> v </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02

EllipticE[z]