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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/07.37.06.0034.01









  


  










Input Form





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










Standard Form





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MathML Form







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</ci> </apply> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <ci> &#977; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <cos /> <ci> &#977; </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> &#977; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> &#977; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> &#955; </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <ci> &#977; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> &#8469; </ci> </apply> <apply> <in /> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <integers /> </apply> <apply> <leq /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> <ci> &#956; </ci> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29