Spherical harmonic function: Series representations (formula 07.37.06.0034)

SphericalHarmonicY

$Mathematica Notation$

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

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mi> Y </mi> <mi> λ </mi> <mi> μ </mi> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> φ </mi> <mo> ⁢ </mo> <mi> μ </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mi> λ </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> - </mo> <mi> μ </mi> <mo> + 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<mrow> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mfrac> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ϑ </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Lambda]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Lambda]", "+", "\[Mu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], 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type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> λ </ci> </apply> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> ϑ </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> μ </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> ϑ </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> μ </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -2 </cn> <ci> λ </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> ϑ </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> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <cos /> <ci> ϑ </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <cos /> <apply> <times /> <ci> ϑ </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> μ </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> ϑ </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> μ </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> λ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> λ </ci> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </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> ϑ </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> λ </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> ℕ </ci> </apply> <apply> <in /> <apply> <plus /> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> <integers /> </apply> <apply> <leq /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> <ci> μ </ci> <apply> <plus /> <ci> λ </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

SphericalHarmonicY[n,m,theta,phi]