Hyperbolic sine: Differentiation (formula 01.19.20.0010)

Sinh

$Mathematica Notation$

Elementary Functions

Sinh[z]

Differentiation

Fractional integro-differentiation

http://functions.wolfram.com/01.19.20.0010.01

Input Form

D[Sinh[z]^n, {z, \[Alpha]}] == (I^n Binomial[n, n/2] (1 - Mod[n, 2]) (1/(z^\[Alpha] Gamma[1 - \[Alpha]])))/ 2^n - (2^(\[Alpha] - n) I^n Sqrt[Pi] Sum[(-1)^(k - 1) Binomial[n, k] (2 Cos[(n Pi)/2] HypergeometricPFQRegularized[{1}, {1/2 - \[Alpha]/2, 1 - \[Alpha]/2}, (1/4) (-2 k + n)^2 z^2] - I (-2 k + n) z HypergeometricPFQRegularized[{1}, {1 - \[Alpha]/2, 3/2 - \[Alpha]/2}, (1/4) (-2 k + n)^2 z^2] Sin[(n Pi)/2]), {k, 0, Floor[(n - 1)/2]}])/z^\[Alpha] /; Element[n, Integers] && n > 0

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)

2002-12-18