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SpheroidalS2




Mathematica Notation

Traditional Notation









Mathieu and Spheroidal Functions > SpheroidalS2[nu,mu,gamma,z] > Series representations > Generalized power series > Expansions at generic point z==z0





http://functions.wolfram.com/11.11.06.0001.01









  


  








Input Form





SpheroidalS2[\[Nu], \[Mu], \[Gamma], z] \[Proportional] SpheroidalS2[\[Nu], \[Mu], \[Gamma], Subscript[z, 0]] + SpheroidalS2Prime[\[Nu], \[Mu], \[Gamma], Subscript[z, 0]] (z - Subscript[z, 0]) + (1/(2 (1 - Subscript[z, 0]^2))) (2 SpheroidalS2Prime[\[Nu], \[Mu], \[Gamma], Subscript[z, 0]] Subscript[z, 0] + SpheroidalS2[\[Nu], \[Mu], \[Gamma], Subscript[z, 0]] (-SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]] + \[Mu]^2/(1 - Subscript[z, 0]^2) + \[Gamma]^2 (-1 + Subscript[z, 0]^2))) (z - Subscript[z, 0])^2 - (1/(6 (-1 + Subscript[z, 0]^2)^3)) (SpheroidalS2Prime[\[Nu], \[Mu], \[Gamma], Subscript[z, 0]] (-1 + Subscript[z, 0]^2) (-2 + \[Gamma]^2 - \[Mu]^2 - 2 (3 + \[Gamma]^2) Subscript[z, 0]^2 + \[Gamma]^2 Subscript[z, 0]^4 - SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]] (-1 + Subscript[z, 0]^2)) - 2 SpheroidalS2[\[Nu], \[Mu], \[Gamma], Subscript[z, 0]] Subscript[z, 0] (-3 \[Mu]^2 - 2 SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]] (-1 + Subscript[z, 0]^2) + \[Gamma]^2 (-1 + Subscript[z, 0]^2)^2)) (z - Subscript[z, 0])^3 + (1/(24 (-1 + Subscript[z, 0]^2)^4)) (4 SpheroidalS2Prime[\[Nu], \[Mu], \[Gamma], Subscript[z, 0]] Subscript[z, 0] (-1 + Subscript[z, 0]^2) (-6 + \[Gamma]^2 - 3 \[Mu]^2 - 2 (3 + \[Gamma]^2) Subscript[z, 0]^2 + \[Gamma]^2 Subscript[z, 0]^4 - 2 SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]] (-1 + Subscript[z, 0]^2)) + SpheroidalS2[\[Nu], \[Mu], \[Gamma], Subscript[z, 0]] (SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]]^2 (-1 + Subscript[z, 0]^2)^2 + \[Gamma]^4 (-1 + Subscript[z, 0]^2)^4 - 2 \[Gamma]^2 (-1 + Subscript[z, 0]^2)^2 (2 + \[Mu]^2 + 4 Subscript[z, 0]^2) + \[Mu]^2 (8 + \[Mu]^2 + 36 Subscript[z, 0]^2) - 2 SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]] (-1 + Subscript[z, 0]^2) (-3 + \[Gamma]^2 - \[Mu]^2 - (9 + 2 \[Gamma]^2) Subscript[z, 0]^2 + \[Gamma]^2 Subscript[z, 0]^4))) (z - Subscript[z, 0])^4 + \[Ellipsis] /; (z -> Subscript[z, 0])








Standard Form





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MathML Form







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</mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <semantics> <mrow> <msub> <mi> &#955; </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> </msub> <mo> ( </mo> <mi> &#947; </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[&quot;\[Lambda]&quot;, RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;(&quot;, TagBox[&quot;\[Gamma]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalEigenvalue[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> S </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> , </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[&quot;S&quot;, RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalS2, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalS2, Rule[Editable, True], Rule[Selectable, True]]]], RowBox[List[&quot;(&quot;, &quot;2&quot;, &quot;)&quot;]]], &quot;(&quot;, RowBox[List[TagBox[&quot;\[Gamma]&quot;, SpheroidalS2, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], SpheroidalS2, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalS2[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mrow> <msub> <mi> &#955; </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> </msub> <mo> ( </mo> <mi> &#947; </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[&quot;\[Lambda]&quot;, RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;(&quot;, TagBox[&quot;\[Gamma]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalEigenvalue[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <semantics> <mrow> <msup> <msubsup> <mi> S </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> , </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SuperscriptBox[SubsuperscriptBox[&quot;S&quot;, RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]]]], RowBox[List[&quot;(&quot;, &quot;2&quot;, &quot;)&quot;]]], &quot;\[Prime]&quot;], &quot;(&quot;, RowBox[List[TagBox[&quot;\[Gamma]&quot;, SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalS2Prime[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mrow> <msub> <mi> &#955; </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> </msub> <mo> ( </mo> <mi> &#947; </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[&quot;\[Lambda]&quot;, RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;(&quot;, TagBox[&quot;\[Gamma]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalEigenvalue[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <semantics> <mrow> <msubsup> <mi> S </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> , </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[&quot;S&quot;, RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalS2, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalS2, Rule[Editable, True], Rule[Selectable, True]]]], RowBox[List[&quot;(&quot;, &quot;2&quot;, &quot;)&quot;]]], &quot;(&quot;, RowBox[List[TagBox[&quot;\[Gamma]&quot;, SpheroidalS2, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], SpheroidalS2, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalS2[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#947; </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <semantics> <mrow> <msub> <mi> &#955; </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> </msub> <mo> ( </mo> <mi> &#947; </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[&quot;\[Lambda]&quot;, RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;(&quot;, TagBox[&quot;\[Gamma]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalEigenvalue[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mrow> <msub> <mi> &#955; </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> </msub> <mo> ( </mo> <mi> &#947; </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[&quot;\[Lambda]&quot;, RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;(&quot;, TagBox[&quot;\[Gamma]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalEigenvalue[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 36 </mn> <mo> &#8290; 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Rule Form





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





2007-05-02