Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











SpheroidalS1Prime






Mathematica Notation

Traditional Notation









Mathieu and Spheroidal Functions > SpheroidalS1Prime[nu,mu,gamma,z] > Differential equations > Ordinary linear differential equations and wronskians > For the direct function itself





http://functions.wolfram.com/11.14.13.0009.01









  


  










Input Form





(1 - a^2 r^(2 z)) Derivative[2][w][z] + (-Log[r] - a^2 r^(2 z) Log[r] - 2 Log[s] + 2 a^2 r^(2 z) Log[s]) Derivative[1][w][z] + (a^2 r^(2 z) Log[r]^2 SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]] + (1/(-1 + a^2 r^(2 z))) ((-a^2) r^(2 z) (-\[Mu]^2 + (-1 + a^2 r^(2 z))^2 \[Gamma]^2) Log[r]^2 + (-1 + a^4 r^(4 z)) Log[r] Log[s] - (-1 + a^2 r^(2 z))^2 Log[s]^2)) w[z] == 0 /; w[z] == Subscript[c, 1] s^z SpheroidalS1Prime[\[Nu], \[Mu], \[Gamma], a r^z] + Subscript[c, 2] s^z SpheroidalS2Prime[\[Nu], \[Mu], \[Gamma], a r^z]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]], ")"]], RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "r", "]"]]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]], " ", RowBox[List["Log", "[", "r", "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", "s", "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]], " ", RowBox[List["Log", "[", "s", "]"]]]]]], ")"]], RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]], " ", SuperscriptBox[RowBox[List["Log", "[", "r", "]"]], "2"], RowBox[List["SpheroidalEigenvalue", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]"]], "]"]]]], "+", FractionBox["1", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Mu]", "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]], ")"]], "2"], " ", SuperscriptBox["\[Gamma]", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Log", "[", "r", "]"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["r", RowBox[List["4", " ", "z"]]]]]]], ")"]], " ", RowBox[List["Log", "[", "r", "]"]], " ", RowBox[List["Log", "[", "s", "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "s", "]"]], "2"]]]]], ")"]]]], ")"]], RowBox[List["w", "[", "z", "]"]]]]]], "\[Equal]", "0"]], "/;", " ", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], SuperscriptBox["s", "z"], " ", RowBox[List["SpheroidalS1Prime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], SuperscriptBox["s", "z"], " ", RowBox[List["SpheroidalS2Prime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> w </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> w </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <msub> <mi> &#955; </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> </msub> <mo> ( </mo> <mi> &#947; </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[&quot;\[Lambda]&quot;, RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;(&quot;, TagBox[&quot;\[Gamma]&quot;, SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalEigenvalue[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#947; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> a </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <msup> <mi> s </mi> <mi> z </mi> </msup> <mo> &#8290; </mo> <semantics> <mrow> <msup> <msubsup> <mi> S </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> , </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> r </mi> <mi> z </mi> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SuperscriptBox[SubsuperscriptBox[&quot;S&quot;, RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalS1Prime, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalS1Prime, Rule[Editable, True], Rule[Selectable, True]]]], RowBox[List[&quot;(&quot;, &quot;1&quot;, &quot;)&quot;]]], &quot;\[Prime]&quot;], &quot;(&quot;, RowBox[List[TagBox[&quot;\[Gamma]&quot;, SpheroidalS1Prime, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;a&quot;, &quot; &quot;, SuperscriptBox[&quot;r&quot;, &quot;z&quot;]]], SpheroidalS1Prime, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalS1Prime[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msup> <mi> s </mi> <mi> z </mi> </msup> <mo> &#8290; </mo> <semantics> <mrow> <msup> <msubsup> <mi> S </mi> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> </mrow> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mi> &#947; </mi> <mo> , </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> r </mi> <mi> z </mi> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SuperscriptBox[SubsuperscriptBox[&quot;S&quot;, RowBox[List[TagBox[&quot;\[Nu]&quot;, SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;\[Mu]&quot;, SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]]]], RowBox[List[&quot;(&quot;, &quot;2&quot;, &quot;)&quot;]]], &quot;\[Prime]&quot;], &quot;(&quot;, RowBox[List[TagBox[&quot;\[Gamma]&quot;, SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;a&quot;, &quot; &quot;, SuperscriptBox[&quot;r&quot;, &quot;z&quot;]]], SpheroidalS2Prime, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[SpheroidalS2Prime[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mtext> </mtext> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <times /> <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> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ln /> <ci> r </ci> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ln /> <ci> s </ci> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> r </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <ci> s </ci> </apply> </apply> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> r </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> SpheroidalEigenvalue </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <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> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <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> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> &#947; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ln /> <ci> r </ci> </apply> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <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> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> s </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ln /> <ci> r </ci> </apply> <apply> <ln /> <ci> s </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> w </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> s </ci> <ci> z </ci> </apply> <apply> <ci> SpheroidalS1Prime </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> r </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> s </ci> <ci> z </ci> </apply> <apply> <ci> SpheroidalS2Prime </ci> <ci> &#957; </ci> <ci> &#956; </ci> <ci> &#947; </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> r </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "r_", "]"]]]], "-", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]], " ", RowBox[List["Log", "[", "r_", "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", "s_", "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]], " ", RowBox[List["Log", "[", "s_", "]"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]], " ", SuperscriptBox[RowBox[List["Log", "[", "r_", "]"]], "2"], " ", RowBox[List["SpheroidalEigenvalue", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "\[Gamma]_"]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["a_", "2"]]], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Mu]_", "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]]]]]], ")"]], "2"], " ", SuperscriptBox["\[Gamma]_", "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Log", "[", "r_", "]"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["a_", "4"], " ", SuperscriptBox["r_", RowBox[List["4", " ", "z_"]]]]]]], ")"]], " ", RowBox[List["Log", "[", "r_", "]"]], " ", RowBox[List["Log", "[", "s_", "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]]]]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "s_", "]"]], "2"]]]]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]]]]]]]]], ")"]], " ", RowBox[List["w", "[", "z_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["s", "z"], " ", RowBox[List["SpheroidalS1Prime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["s", "z"], " ", RowBox[List["SpheroidalS2Prime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]]]]]]]]]]]]










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





2007-05-02





© 1998- Wolfram Research, Inc.