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http://functions.wolfram.com/11.14.13.0010.01
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Wronskian[s^z SpheroidalS1Prime[\[Nu], \[Mu], \[Gamma], a r^z],
s^z SpheroidalS2Prime[\[Nu], \[Mu], \[Gamma], a r^z], z] ==
a s^(2 z) r^z Log[r] (-(\[Gamma]^2/(1 - a^2 r^(2 z))) +
\[Mu]^2/(1 - a^2 r^(2 z))^3 - SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]]/
(1 - a^2 r^(2 z))^2) (SpheroidalS1Prime[\[Nu], \[Mu], \[Gamma], 0]
SpheroidalS2[\[Nu], \[Mu], \[Gamma], 0] -
SpheroidalS1[\[Nu], \[Mu], \[Gamma], 0] SpheroidalS2Prime[\[Nu], \[Mu],
\[Gamma], 0])
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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> <msub> <mi> W </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msup> <mi> s </mi> <mi> z </mi> </msup> <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> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> r </mi> <mi> z </mi> </msup> </mrow> </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, 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Rule[Selectable, True]]]], RowBox[List["(", "2", ")"]]], "\[Prime]"], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", " ", SuperscriptBox["r", "z"]]], SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalS2Prime[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> s </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> r </mi> <mi> z </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msup> <mi> γ </mi> <mn> 2 </mn> </msup> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> </mfrac> </mrow> <mo> + </mo> <mfrac> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <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> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <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> <mn> 0 </mn> </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["0", SpheroidalS1Prime, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalS1Prime[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <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> <mrow> <mi> γ </mi> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox["S", RowBox[List[TagBox["\[Nu]", SpheroidalS2, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalS2, Rule[Editable, True], Rule[Selectable, True]]]], RowBox[List["(", "2", ")"]]], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalS2, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["0", SpheroidalS2, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalS2[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <semantics> <mrow> <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> <mrow> <mi> γ </mi> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox["S", RowBox[List[TagBox["\[Nu]", SpheroidalS1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalS1, Rule[Editable, True], Rule[Selectable, True]]]], RowBox[List["(", "1", ")"]]], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalS1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["0", SpheroidalS1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalS1[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <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> <mn> 0 </mn> </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["0", SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalS2Prime[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <ci> s </ci> <ci> z </ci> </apply> <apply> <ci> SpheroidalS1Prime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> r </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> s </ci> <ci> z </ci> </apply> <apply> <ci> SpheroidalS2Prime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> r </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> s </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> r </ci> <ci> z </ci> </apply> <apply> <ln /> <ci> r </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> SpheroidalEigenvalue </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> SpheroidalS1Prime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> SpheroidalS2 </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> SpheroidalS1 </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> SpheroidalS2Prime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <cn type='integer'> 0 </cn> </apply> </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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