Angular spheroidal function of the second kind: Differential equations (formula 11.09.13.0004)

SpheroidalQS

$Mathematica Notation$

Mathieu and Spheroidal Functions

SpheroidalQS[nu,mu,gamma,z]

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

http://functions.wolfram.com/11.09.13.0004.01

Input Form

Wronskian[SpheroidalQS[\[Nu], \[Mu], \[Gamma], g[z]], SpheroidalPS[\[Nu], \[Mu], \[Gamma], g[z]], z] == (Derivative[1][g][z]/(1 - g[z]^2)) (SpheroidalPSPrime[\[Nu], \[Mu], \[Gamma], 0] SpheroidalQS[\[Nu], \[Mu], \[Gamma], 0] - SpheroidalPS[\[Nu], \[Mu], \[Gamma], 0] SpheroidalQSPrime[\[Nu], \[Mu], \[Gamma], 0])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Wronskian", "[", RowBox[List[RowBox[List["SpheroidalQS", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]], ",", RowBox[List["SpheroidalPS", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], RowBox[List["1", "-", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["SpheroidalPSPrime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "0"]], "]"]], " ", RowBox[List["SpheroidalQS", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "0"]], "]"]]]], "-", RowBox[List[RowBox[List["SpheroidalPS", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "0"]], "]"]], " ", RowBox[List["SpheroidalQSPrime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "0"]], "]"]]]]]], ")"]], " "]]]]]]

MathML Form

<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> <semantics> <mrow> <msub> <mi> QS </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> , </mo> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[StyleBox["QS", "IT"], RowBox[List[TagBox["\[Nu]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]]]]], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["g", "(", "z", ")"]], SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalQS[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <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> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </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[RowBox[List["g", "(", "z", ")"]], SpheroidalPS, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalPS[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> g </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mrow> <msup> <msub> <mi> PS </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <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[SubscriptBox[StyleBox["PS", "IT"], RowBox[List[TagBox["\[Nu]", SpheroidalPSPrime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalPSPrime, Rule[Editable, True], Rule[Selectable, True]]]]], "\[Prime]"], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalPSPrime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["0", SpheroidalPSPrime, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalPSPrime[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <msub> <mi> QS </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["QS", "IT"], RowBox[List[TagBox["\[Nu]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]]]]], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["0", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalQS[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <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> <semantics> <mrow> <msup> <msub> <mi> QS </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <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[SubscriptBox[StyleBox["QS", "IT"], RowBox[List[TagBox["\[Nu]", SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]]]]], "\[Prime]"], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["0", SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalQSPrime[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> <ci> SpheroidalQS </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <apply> <ci> SpheroidalPS </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> g </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> SpheroidalPSPrime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> SpheroidalQS </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> SpheroidalPS </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> SpheroidalQSPrime </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>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Wronskian", "[", RowBox[List[RowBox[List["SpheroidalQS", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "\[Gamma]_", ",", RowBox[List["g", "[", "z_", "]"]]]], "]"]], ",", RowBox[List["SpheroidalPS", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "\[Gamma]_", ",", RowBox[List["g", "[", "z_", "]"]]]], "]"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["SpheroidalPSPrime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "0"]], "]"]], " ", RowBox[List["SpheroidalQS", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "0"]], "]"]]]], "-", RowBox[List[RowBox[List["SpheroidalPS", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "0"]], "]"]], " ", RowBox[List["SpheroidalQSPrime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "0"]], "]"]]]]]], ")"]]]], RowBox[List["1", "-", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]]]]]]]

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

2007-05-02