Jacobi elliptic function nc: Differentiation (formula 09.31.20.0006)

JacobiNC

$Mathematica Notation$

Elliptic Functions

JacobiNC[z,m]

Differentiation

Fractional integro-differentiation

With respect to z

http://functions.wolfram.com/09.31.20.0006.01

Input Form

D[JacobiNC[z, m], {z, \[Alpha]}] == (Pi/(2 Sqrt[1 - m] EllipticK[m])) Sum[(((-1)^k EulerE[2 k])/(EllipticK[m]^(2 k) Gamma[2 k - \[Alpha] + 1])) (Pi/2)^(2 k) z^(2 k - \[Alpha]), {k, 0, Infinity}] - ((2^(\[Alpha] + 1) Pi^(3/2))/(z^\[Alpha] (Sqrt[1 - m] EllipticK[m]))) Sum[(((-1)^k EllipticNomeQ[m]^(2 k + 1))/(EllipticNomeQ[m]^(2 k + 1) + 1)) HypergeometricPFQRegularized[{1}, {(1 - \[Alpha])/2, 1 - \[Alpha]/2}, -(((2 k + 1)^2 Pi^2 z^2)/(16 EllipticK[m]^2))], {k, 0, Infinity}]

Standard Form

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MathML Form

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Rule Form

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

2001-10-29