Spherical harmonic function: Series representations (formula 07.37.06.0028)

SphericalHarmonicY

$Mathematica Notation$

Hypergeometric Functions

SphericalHarmonicY[lambda,mu,theta,phi]

Series representations

Generalized power series

Expansions at cos(theta)==infinity

http://functions.wolfram.com/07.37.06.0028.01

Input Form

SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]] == ((Sqrt[2 \[Lambda] + 1] Sqrt[Gamma[\[Lambda] - \[Mu] + 1]])/ (2 Pi Sqrt[Gamma[\[Lambda] + \[Mu] + 1]])) E^(I \[CurlyPhi] \[Mu]) ((Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2)/(Sin[\[CurlyTheta]/2]^2)^(\[Mu]/2)) (((2^\[Lambda] (Cos[\[CurlyTheta]] - 1)^\[Lambda])/ Gamma[\[Lambda] - \[Mu] + 1]) Gamma[1/2 + \[Lambda]] HypergeometricPFQ[{\[Mu] - \[Lambda], -\[Lambda]}, {-2 \[Lambda]}, 2/(1 - Cos[\[CurlyTheta]])] + ((2^(-1 - \[Lambda]) (Cos[\[CurlyTheta]] - 1)^(-1 - \[Lambda]))/ Gamma[-\[Mu] - \[Lambda]]) Gamma[-(1/2) - \[Lambda]] HypergeometricPFQ[{1 + \[Lambda], 1 + \[Mu] + \[Lambda]}, {2 + 2 \[Lambda]}, 2/(1 - Cos[\[CurlyTheta]])]) /; !Element[\[CurlyTheta], Reals] && !Element[2 \[Lambda], Integers]

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]