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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Hypergeometric Functions > SphericalHarmonicY[lambda,mu,theta,phi] > Representations through more general functions > Through hypergeometric functions > Involving 2F1





http://functions.wolfram.com/07.37.26.0004.01









  


  










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]) (1/Sqrt[Pi]) ((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[-m - \[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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</ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <notin /> <ci> &#977; </ci> <ci> &#8477; </ci> </apply> <apply> <notin /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2001-10-29





© 1998- Wolfram Research, Inc.