Spherical harmonic function: Differentiation (formula 07.37.20.0003)

SphericalHarmonicY

$Mathematica Notation$

Hypergeometric Functions

SphericalHarmonicY[lambda,mu,theta,phi]

Differentiation

Low-order differentiation

With respect to lambda

http://functions.wolfram.com/07.37.20.0003.01

Input Form

D[SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]], \[Lambda]] == (-((2 \[Lambda] + 1)/(2 Gamma[2 - \[Mu]]))) Sqrt[(2 \[Lambda] + 1)/Pi] (Sqrt[Gamma[\[Lambda] - \[Mu] + 1]]/ Sqrt[Gamma[\[Lambda] + \[Mu] + 1]]) E^(I \[Mu] \[CurlyPhi]) ((Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2)/(Sin[\[CurlyTheta]/2]^2)^ (\[Mu]/2 - 1)) HypergeometricPFQ[{{1 - \[Lambda], 2 + \[Lambda]}, {1}, {1, -\[Lambda], 1 + \[Lambda]}}, {{2, 2 - \[Mu]}, {}, {2 + \[Lambda], 1 - \[Lambda]}}, Sin[\[CurlyTheta]/2]^2, Sin[\[CurlyTheta]/2]^2] + (1/(2 (1 + 2 \[Lambda]))) (2 + (1 + 2 \[Lambda]) (PolyGamma[1 + \[Lambda] - \[Mu]] - PolyGamma[1 + \[Lambda] + \[Mu]])) SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]]

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

SphericalHarmonicY[n,m,theta,phi]