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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Series representations > Generalized power series > Expansions at cos(theta)==infinity





http://functions.wolfram.com/07.37.06.0030.01









  


  










Input Form





SphericalHarmonicY[\[Lambda], \[Mu], \[CurlyTheta], \[CurlyPhi]] == E^(I \[CurlyPhi] \[Mu]) ((Sin[Pi (\[Lambda] - \[Mu])] Sin[Pi \[Lambda]] Sqrt[2 \[Lambda] + 1] Sqrt[Gamma[\[Lambda] - \[Mu] + 1]] Sqrt[Gamma[\[Lambda] + \[Mu] + 1]])/(2^(\[Lambda] + 1) Pi^2 Gamma[3/2 + \[Lambda]])) (Cos[\[CurlyTheta]] - 1)^(-\[Lambda] - 1) ((Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2)/(Sin[\[CurlyTheta]/2]^2)^(\[Mu]/2)) Sum[((Pochhammer[\[Lambda] + 1, k] Pochhammer[\[Lambda] + \[Mu] + 1, k])/ (k! Pochhammer[2 \[Lambda] + 2, k])) (PolyGamma[k + 1] - PolyGamma[-k - \[Lambda] - \[Mu]] - PolyGamma[\[Lambda] + k + 1] + PolyGamma[2 \[Lambda] + k + 2]) (2/(1 - Cos[\[CurlyTheta]]))^k, {k, 0, Infinity}] + ((2^(\[Lambda] + 1) Sin[Pi (\[Mu] - \[Lambda])] Gamma[\[Lambda] + \[Mu] + 1])/(Pi Gamma[-\[Lambda]] Gamma[2 \[Lambda] + 2])) (Cos[\[CurlyTheta]] - 1)^(-\[Lambda] - 1) ((Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2)/(Sin[\[CurlyTheta]/2]^2)^(\[Mu]/2)) Log[(Cos[\[CurlyTheta]] - 1)/2] Hypergeometric2F1[\[Lambda] + 1, \[Lambda] + \[Mu] + 1, 2 \[Lambda] + 2, 2/(1 - Cos[\[CurlyTheta]])] + (1/(2^\[Lambda] Gamma[\[Lambda] + 1])) (Cos[\[CurlyTheta]] - 1)^\[Lambda] ((Cos[\[CurlyTheta]/2]^2)^(\[Mu]/2)/(Sin[\[CurlyTheta]/2]^2)^(\[Mu]/2)) Sum[(((2 \[Lambda] - k)! Pochhammer[-\[Lambda], k])/ (k! Gamma[1 - k - \[Mu] + \[Lambda]])) (2/(1 - Cos[\[CurlyTheta]]))^k, {k, 0, 2 \[Lambda]}] /; Element[2 \[Lambda] + 1, Integers] && 2 \[Lambda] + 1 >= 0 && !Element[\[Lambda] - \[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





© 1998- Wolfram Research, Inc.