Eigenvalue of the spheroidal wave functions: Specific values (formula 11.16.03.0003)

SpheroidalEigenvalue

$Mathematica Notation$

Mathieu and Spheroidal Functions

SpheroidalEigenvalue[nu,mu,gamma]

Specific values

Specialized values

For fixed nu, lambda

http://functions.wolfram.com/11.16.03.0003.01

Input Form

SpheroidalEigenvalue[\[Nu], 1/2, \[Gamma]] == -(1/4) - \[Gamma]^2/2 + MathieuCharacteristicA[1/2 + \[Nu], \[Gamma]^2/4]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["SpheroidalEigenvalue", "[", RowBox[List["\[Nu]", ",", FractionBox["1", "2"], ",", "\[Gamma]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "-", FractionBox[SuperscriptBox["\[Gamma]", "2"], "2"], "+", RowBox[List["MathieuCharacteristicA", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", FractionBox[SuperscriptBox["\[Gamma]", "2"], "4"]]], "]"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <msub> <mi> λ </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </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[FractionBox["1", "2"], 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> <mrow> <mrow> <mo> - </mo> <mfrac> <msup> <mi> γ </mi> <mn> 2 </mn> </msup> <mn> 2 </mn> </mfrac> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> + </mo> <mrow> <msub> <semantics> <mi> a </mi> <annotation-xml encoding='MathML-Content'> <ci> MathieuCharacteristicA </ci> </annotation-xml> </semantics> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mfrac> <msup> <mi> γ </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> SpheroidalEigenvalue </ci> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> γ </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 /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <ci> MathieuCharacteristicA </ci> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SpheroidalEigenvalue", "[", RowBox[List["\[Nu]_", ",", FractionBox["1", "2"], ",", "\[Gamma]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "-", FractionBox[SuperscriptBox["\[Gamma]", "2"], "2"], "+", RowBox[List["MathieuCharacteristicA", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", FractionBox[SuperscriptBox["\[Gamma]", "2"], "4"]]], "]"]]]]]]]]

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

2007-05-02