|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/11.15.20.0001.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
D[SpheroidalS2Prime[\[Nu], \[Mu], \[Gamma], z], z] ==
(1/(1 - z^2)) ((-((1 - z^2) \[Gamma]^2 - \[Mu]^2/(1 - z^2) +
SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]]))
SpheroidalS2[\[Nu], \[Mu], \[Gamma], z] +
2 z SpheroidalS2Prime[\[Nu], \[Mu], \[Gamma], z])
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "z"], RowBox[List["SpheroidalS2Prime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], " ", SuperscriptBox["\[Gamma]", "2"]]], "-", FractionBox[SuperscriptBox["\[Mu]", "2"], RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], "+", RowBox[List["SpheroidalEigenvalue", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]"]], "]"]]]], ")"]]]], " ", RowBox[List["SpheroidalS2", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "z"]], "]"]]]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["SpheroidalS2Prime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "z"]], "]"]]]]]], ")"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <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> </mrow> <mrow> <mo> ∂ </mo> <mi> z </mi> </mrow> </mfrac> <mo>  </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> <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> </mrow> <mo> - </mo> <mrow> <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> <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> </mrow> <mo> ) </mo> </mrow> <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> <mi> z </mi> </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["z", SpheroidalS2, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalS2[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> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> SpheroidalS2Prime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <ci> z </ci> </apply> </apply> <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'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <ci> SpheroidalS2Prime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <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> <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> </apply> <apply> <ci> SpheroidalS2 </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], RowBox[List["SpheroidalS2Prime", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "\[Gamma]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], " ", SuperscriptBox["\[Gamma]", "2"]]], "-", FractionBox[SuperscriptBox["\[Mu]", "2"], RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], "+", RowBox[List["SpheroidalEigenvalue", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]"]], "]"]]]], ")"]]]], " ", RowBox[List["SpheroidalS2", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "z"]], "]"]]]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["SpheroidalS2Prime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "z"]], "]"]]]]]], RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
){kind=small}
