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http://functions.wolfram.com/11.14.13.0001.01
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(1 - z^2) Derivative[2][w][z] -
2 z (2 + (\[Mu]^2 + (1 - z^2)^2 \[Gamma]^2)/
(\[Mu]^2 - (1 - z^2) (SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]] +
(1 - z^2) \[Gamma]^2))) Derivative[1][w][z] +
(-(\[Mu]^2/(1 - z^2)) + (1 - z^2) \[Gamma]^2 - 2 +
SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]] +
(4 z^2 (\[Mu]^2 + (1 - z^2)^2 \[Gamma]^2))/
((1 - z^2) (\[Mu]^2 - (1 - z^2)^2 \[Gamma]^2 -
(1 - z^2) SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]]))) w[z] ==
0 /; w[z] == Subscript[c, 1] SpheroidalS1Prime[\[Nu], \[Mu], \[Gamma], z] +
Subscript[c, 2] SpheroidalS2Prime[\[Nu], \[Mu], \[Gamma], z]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> w </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> γ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> γ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <semantics> <mrow> <msub> <mi> λ </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mi> γ </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox["\[Lambda]", RowBox[List[TagBox["\[Nu]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]]]]], "(", TagBox["\[Gamma]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[SpheroidalEigenvalue[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> w </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> γ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ⁢ </mo> <msup> <mi> γ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <msub> <mi> λ </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mi> γ </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox["\[Lambda]", RowBox[List[TagBox["\[Nu]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]]]]], "(", TagBox["\[Gamma]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[SpheroidalEigenvalue[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> γ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <semantics> <mrow> <msub> <mi> λ </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mi> γ </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox["\[Lambda]", RowBox[List[TagBox["\[Nu]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]]]]], "(", TagBox["\[Gamma]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[SpheroidalEigenvalue[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> - </mo> <mfrac> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo>  </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <semantics> <mrow> <msup> <msubsup> <mi> S </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SuperscriptBox[SubsuperscriptBox["S", RowBox[List[TagBox["\[Nu]", SpheroidalS1Prime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalS1Prime, Rule[Editable, True], Rule[Selectable, True]]]], RowBox[List["(", "1", ")"]]], "\[Prime]"], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalS1Prime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["z", SpheroidalS1Prime, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalS1Prime[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <semantics> <mrow> <msup> <msubsup> <mi> S </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SuperscriptBox[SubsuperscriptBox["S", RowBox[List[TagBox["\[Nu]", SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]]]], RowBox[List["(", "2", ")"]]], "\[Prime]"], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["z", SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalS2Prime[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mtext> </mtext> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> SpheroidalEigenvalue </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> SpheroidalEigenvalue </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> SpheroidalEigenvalue </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -2 </cn> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> w </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> SpheroidalS1Prime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> SpheroidalS2Prime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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