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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/07.37.06.0036.01









  


  










Input Form





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










Standard Form





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MathML Form







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</mo> <mi> &#966; </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> </msup> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mi> &#955; </mi> </msup> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mfrac> <msup> <mrow> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> &#977; </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> &#977; </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> - </mo> <mi> &#955; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Lambda]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Lambda]&quot;, &quot;+&quot;, &quot;\[Mu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> &#955; </mi> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> &#955; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> &#955; </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> &#955; </mi> </msup> <mo> &#8290; </mo> <mfrac> <msup> <mrow> <msup> <mi> cos </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> &#977; </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mfrac> <mi> &#977; </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Mu]&quot;, &quot;-&quot;, &quot;\[Lambda]&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;\[Lambda]&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;2&quot;]], &quot; &quot;, &quot;\[Lambda]&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> </mrow> </msup> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#955; </mi> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; 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Rule Form





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





2001-10-29





© 1998-2014 Wolfram Research, Inc.