Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

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
 











variants of this functions
GegenbauerC




Mathematica Notation

Traditional Notation









Hypergeometric Functions > GegenbauerC[nu,lambda,z] > Differential equations > Ordinary linear differential equations and wronskians > For the direct function itself





http://functions.wolfram.com/07.14.13.0004.01









  


  








Input Form





Wronskian[GegenbauerC[\[Nu], \[Lambda], g[z]], (1 - g[z]^2)^((1/4) (1 - 2 \[Lambda])) LegendreQ[\[Nu] + \[Lambda] - 1/2, 1/2 - \[Lambda], 2, g[z]], z] == ((2^(1/2 - \[Lambda]) Sqrt[Pi] (1 - g[z]^2)^(-(1/2) - \[Lambda]))/ Gamma[\[Lambda]]) Derivative[1][g][z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Wronskian", "[", RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]], ")"]], RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]]]], ")"]]]]], " ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]", "+", "\[Lambda]", "-", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox["1", "2"], " ", "-", "\[Lambda]"]], ",", "2", ",", RowBox[List["g", "[", "z", "]"]]]], "]"]]]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]]]]], RowBox[List["Gamma", "[", "\[Lambda]", "]"]]], RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "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> <msub> <mi> W </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msubsup> <mi> C </mi> <mi> &#957; </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> </mrow> <mn> 4 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox[&quot;Q&quot;, LegendreQ] </annotation> </semantics> <mrow> <mi> &#955; </mi> <mo> + </mo> <mi> &#957; </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> &#955; </mi> </mrow> </msubsup> <mo> ( </mo> <semantics> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;g&quot;, &quot;(&quot;, &quot;z&quot;, &quot;)&quot;]], HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> &#955; </mi> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> g </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#955; </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <ci> z </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> &#957; </ci> </apply> <ci> &#955; </ci> </apply> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <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> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> LegendreQ </ci> <apply> <plus /> <ci> &#955; </ci> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> </apply> <cn type='integer'> 2 </cn> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </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> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> &#955; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Wronskian", "[", RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", RowBox[List["g", "[", "z_", "]"]]]], "]"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["g", "[", "z_", "]"]], "2"]]], ")"]], RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]_"]]]], ")"]]]]], " ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]_", "+", "\[Lambda]_", "-", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]_"]], ",", "2", ",", RowBox[List["g", "[", "z_", "]"]]]], "]"]]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["g", "[", "z", "]"]], "2"]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]]]]], ")"]], " ", RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]]


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





2007-05-02