Derivative of the angular spheroidal function of the first kind: Series representations (formula 11.12.06.0005)

SpheroidalPSPrime

$Mathematica Notation$

Mathieu and Spheroidal Functions

SpheroidalPSPrime[nu,mu,gamma,z]

Series representations

Generalized power series

Expansions at z==0

http://functions.wolfram.com/11.12.06.0005.01

Input Form

SpheroidalPSPrime[\[Nu], \[Mu], \[Gamma], z] \[Proportional] SpheroidalPSPrime[\[Nu], \[Mu], \[Gamma], 0] - (\[Gamma]^2 - \[Mu]^2 + SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]]) SpheroidalPS[\[Nu], \[Mu], \[Gamma], 0] z + (-(1/2)) (-2 + \[Gamma]^2 - \[Mu]^2 + SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]]) SpheroidalPSPrime[\[Nu], \[Mu], \[Gamma], 0] z^2 + (1/6) (\[Gamma]^4 - 2 \[Gamma]^2 (2 + \[Mu]^2) + \[Mu]^2 (8 + \[Mu]^2) + 2 (-3 + \[Gamma]^2 - \[Mu]^2) SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]] + SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]]^2) SpheroidalPS[\[Nu], \[Mu], \[Gamma], 0] z^3 + \[Ellipsis] /; (z -> 0)

Standard Form

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MathML Form

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</mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 3 </mn> </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> <mo> ⁢ </mo> <semantics> <mrow> <msub> <mi> PS </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[StyleBox["PS", "IT"], RowBox[List[TagBox["\[Nu]", SpheroidalPS, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalPS, Rule[Editable, True], Rule[Selectable, True]]]]], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalPS, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["0", SpheroidalPS, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalPS[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mo> … </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> SpheroidalPSPrime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> SpheroidalPSPrime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> SpheroidalEigenvalue </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> </apply> </apply> <apply> <ci> SpheroidalPS </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <cn type='integer'> 0 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> SpheroidalEigenvalue </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> </apply> <cn type='integer'> -2 </cn> </apply> <apply> <ci> SpheroidalPSPrime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <apply> <power /> <ci> γ </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> SpheroidalEigenvalue </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -3 </cn> </apply> <apply> <ci> SpheroidalEigenvalue </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> </apply> </apply> </apply> <apply> <ci> SpheroidalPS </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <ci> … </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02