Spherical harmonic function: Representations through more general functions (formula 07.37.26.0003)

SphericalHarmonicY

$Mathematica Notation$

Hypergeometric Functions

SphericalHarmonicY[lambda,mu,theta,phi]

Representations through more general functions

Through hypergeometric functions

Involving ₂F₁

http://functions.wolfram.com/07.37.26.0003.01

Input Form

SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]] == Sqrt[(2 \[Lambda] + 1)/(4 Pi)] (Sqrt[Gamma[\[Lambda] - \[Mu] + 1]]/ Sqrt[Gamma[\[Lambda] + \[Mu] + 1]]) E^(I \[CurlyPhi] \[Mu]) ((Gamma[-\[Mu]]/(Gamma[-\[Lambda] - \[Mu]] Gamma[\[Lambda] - \[Mu] + 1])) ((Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2)/(Sin[\[CurlyTheta]/2]^2)^(\[Mu]/2)) Hypergeometric2F1[-\[Lambda], \[Lambda] + 1, 1 + \[Mu], Cos[\[CurlyTheta]/2]^2] - ((Sin[Pi \[Lambda]] Gamma[\[Mu]])/Pi) (1/((Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2) (Sin[\[CurlyTheta]/2]^2)^ (\[Mu]/2))) Hypergeometric2F1[\[Lambda] - \[Mu] + 1, -\[Lambda] - \[Mu], 1 - \[Mu], Cos[\[CurlyTheta]/2]^2]) /; !Element[\[Mu], 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]