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variants of this functions
SphericalHarmonicY




Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Representations through more general functions > Through hypergeometric functions > Involving 2F1





http://functions.wolfram.com/05.10.26.0004.01









  


  








Input Form





SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] == ((2^(n - 1) Sqrt[2 n + 1] Gamma[1/2 + n])/(Pi Sqrt[Gamma[n - m + 1]] Sqrt[Gamma[n + m + 1]])) E^(I \[CurlyPhi] m) ((Cos[\[CurlyTheta]/2]^2)^(m/2)/(Sin[\[CurlyTheta]/2]^2)^(m/2)) (Cos[\[CurlyTheta]] - 1)^n Hypergeometric2F1[-n, m - n, -2 n, 2/(1 - Cos[\[CurlyTheta]])]








Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <mi> Y </mi> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> , </mo> <mi> &#966; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#966; </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msup> </mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8289; 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Rule Form





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





2001-10-29