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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 theta==Pi





http://functions.wolfram.com/07.37.06.0024.01









  


  








Input Form





SphericalHarmonicY[\[Lambda], 0, \[CurlyTheta], \[CurlyPhi]] \[Proportional] (Sin[Pi \[Lambda]]/2) Sqrt[(2 \[Lambda] + 1)/Pi^3] (Log[(\[CurlyTheta] - Pi)^2/4] (1 - ((\[Lambda] (\[Lambda] + 1))/4) (\[CurlyTheta] - Pi)^2 + ((\[Lambda] (-2 + \[Lambda] + 6 \[Lambda]^2 + 3 \[Lambda]^3))/192) (\[CurlyTheta] - Pi)^4 + O[(\[CurlyTheta] - Pi)^6]) + 2 EulerGamma + PolyGamma[-\[Lambda]] + PolyGamma[\[Lambda] + 1] - (1/12 + ((\[Lambda] (\[Lambda] + 1))/4) (-2 + 2 EulerGamma + PolyGamma[1 - \[Lambda]] + PolyGamma[2 + \[Lambda]])) (\[CurlyTheta] - Pi)^2 - ((1/2880) (2 - 60 \[Lambda] (\[Lambda] + 1) - 60 \[Lambda] (\[Lambda] + 1) (-2 + 2 EulerGamma + PolyGamma[1 - \[Lambda]] + PolyGamma[\[Lambda] + 2]) - 45 (\[Lambda] - 1) \[Lambda] (\[Lambda] + 1) (\[Lambda] + 2) (-3 + 2 EulerGamma + PolyGamma[2 - \[Lambda]] + PolyGamma[\[Lambda] + 3]))) (\[CurlyTheta] - Pi)^4 + O[(\[CurlyTheta] - Pi)^6]) /; (\[CurlyTheta] -> Pi)








Standard Form





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MathML Form







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</mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2880 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 60 </mn> </mrow> <mo> &#8290; </mo> <mi> &#955; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 60 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#955; </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> &#955; 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</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> &#955; </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#977; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> SphericalHarmonicY </ci> <ci> &#955; </ci> <cn type='integer'> 0 </cn> <ci> &#977; </ci> <ci> &#966; </ci> </apply> <apply> <times /> <apply> <times /> <apply> <sin /> <apply> <times /> <pi /> <ci> &#955; </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ln /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> &#977; 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</ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> &#977; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> &#977; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <cn type='integer'> 6 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> &#977; </ci> <pi /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29