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variants of this functions
SphericalHarmonicY






Mathematica Notation

Traditional Notation









Polynomials > SphericalHarmonicY[n,m,theta,phi] > Series representations > Generalized power series > Expansions at theta==0





http://functions.wolfram.com/05.10.06.0019.01









  


  










Input Form





SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]] \[Proportional] (Sqrt[(2 n + 1)/(4 Pi)] (Sqrt[Gamma[n - m + 1]]/Sqrt[Gamma[n + m + 1]]) 2^m E^(I \[CurlyPhi] m) (1/Gamma[1 - m] + (1/12) (-((3 n (n + 1))/Gamma[2 - m]) + 1/Gamma[-m]) \[CurlyTheta]^2 + (1/1440) ((30 n (n + 1) (m + 1))/Gamma[2 - m] + (45 (n - 1) n (n + 1) (n + 2))/Gamma[3 - m] + (7 - 5 m)/Gamma[-m]) \[CurlyTheta]^4 + (1/362880) (-((63 n (n + 1) (4 + m (3 + 5 m)))/ Gamma[2 - m]) - (945 (n - 1) n (n + 1) (n + 2) (m + 2))/ Gamma[3 - m] - (945 (n - 1) n (n + 1) (n + 2) (n^2 + n - 6))/ Gamma[4 - m] + (124 + 7 m (5 m - 21))/Gamma[-m]) \[CurlyTheta]^6 + O[\[CurlyTheta]^8]))/(\[CurlyTheta]^2)^(m/2) /; (\[CurlyTheta] -> 0)










Standard Form





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MathML Form







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</ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 1440 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 30 </cn> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 7 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#977; </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 362880 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 63 </cn> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <cn type='integer'> 4 </cn> <apply> <times /> <ci> m </ci> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 945 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 945 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> n </ci> <cn type='integer'> -6 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 124 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <ci> m </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> m </ci> </apply> <cn type='integer'> -21 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#977; </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> &#977; </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> &#977; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





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