Incomplete elliptic integral of the first kind: Differentiation (formula 08.05.20.0009)

EllipticF

$Mathematica Notation$

Elliptic Integrals

EllipticF[z,m]

Differentiation

Fractional integro-differentiation

With respect to z

http://functions.wolfram.com/08.05.20.0009.01

Input Form

D[EllipticF[z, m], {z, \[Alpha]}] == Sqrt[Pi] z^(1 - \[Alpha]) 2^(\[Alpha] - 1) Sum[(((-1)^(j + l) 2^(-2 j - 2 l) m^l)/ ((1 + 2 j + 2 l) j! l!)) Pochhammer[1/2, j] Pochhammer[1/2, l] Sum[(-1)^p Binomial[1 + 2 j + 2 l, p] (1 + 2 j + 2 l - 2 p) HypergeometricPFQRegularized[{1}, {1 - \[Alpha]/2, (3 - \[Alpha])/2}, -(((1 + 2 j + 2 l - 2 p)^2 z^2)/4)], {p, 0, j + l}], {j, 0, Infinity}, {l, 0, Infinity}]

Standard Form

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MathML Form

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</mrow> <mrow> <mi> j </mi> <mo> + </mo> <mi> l </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> l </mi> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> m </mi> <mi> l </mi> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> l </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> l </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "j"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> 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Rule Form

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

2001-10-29