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
 











variants of this functions
GegenbauerC






Mathematica Notation

Traditional Notation









Hypergeometric Functions > GegenbauerC[nu,lambda,z] > Series representations > Generalized power series > Expansions at z==infinity > For the function itself > Generic formulas for main term





http://functions.wolfram.com/07.14.06.0081.01









  


  










Input Form





GegenbauerC[\[Nu], \[Lambda], z] \[Proportional] Piecewise[{{0, (Element[-\[Nu], Integers] && -\[Nu] > 0) || (Element[-(1/2) - \[Lambda], Integers] && -(1/2) - \[Lambda] >= 0 && ((Element[-2 \[Lambda] - \[Nu], Integers] && -2 \[Lambda] - \[Nu] >= 0 && \[Lambda] + \[Nu] >= 1/2) || (Element[\[Nu], Integers] && \[Nu] >= 0 && \[Lambda] + \[Nu] <= -(1/2))))}, {ComplexInfinity, Element[-(1/2) - \[Lambda], Integers] && -(1/2) - \[Lambda] >= 0 && Element[-2 \[Lambda] - \[Nu], Integers] && -2 \[Lambda] - \[Nu] >= 0 && \[Lambda] + \[Nu] < 1/2}, {(2^\[Nu] z^\[Nu] Gamma[\[Lambda] + \[Nu]])/(Gamma[\[Lambda]] Gamma[1 + \[Nu]]), Re[\[Lambda] + \[Nu]] > 0}, {(-((2^(\[Nu] + 1) Sin[Pi \[Nu]])/Pi)) Log[z] z^\[Nu], \[Lambda] + \[Nu] == 0}, {-((2^(-2 \[Lambda] - \[Nu]) Gamma[-\[Lambda] - \[Nu]] Gamma[2 \[Lambda] + \[Nu]] Sin[Pi \[Nu]] z^(-2 \[Lambda] - \[Nu]))/ (Pi Gamma[\[Lambda]])), Re[\[Lambda] + \[Nu]] < 0}}, -((2^(-2 \[Lambda] - \[Nu]) Gamma[-\[Lambda] - \[Nu]] Gamma[2 \[Lambda] + \[Nu]] Sin[Pi \[Nu]] z^(-2 \[Lambda] - \[Nu]))/ (Pi Gamma[\[Lambda]])) + (2^\[Nu] Gamma[\[Lambda] + \[Nu]] z^\[Nu])/ (Gamma[\[Lambda]] Gamma[1 + \[Nu]])] /; (Abs[z] -> Infinity)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "\[Nu]"]], ">", "0"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]"]], "\[GreaterEqual]", FractionBox["1", "2"]]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]"]], "\[LessEqual]", RowBox[List["-", FractionBox["1", "2"]]]]]]], ")"]]]], ")"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["ComplexInfinity", " ", ",", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]"]], "<", FractionBox["1", "2"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]], ",", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Lambda]", "+", "\[Nu]"]], "]"]], ">", "0"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "+", "1"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "\[Pi]"]]], " ", RowBox[List["Log", "[", "z", "]"]], SuperscriptBox["z", "\[Nu]"]]], " ", ",", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]"]], "\[Equal]", "0"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], ",", RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Lambda]", "+", "\[Nu]"]], "]"]], "<", "0"]]]], "}"]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Nu]"]], "]"]], " ", SuperscriptBox["z", "\[Nu]"]]], RowBox[List[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]]]]]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]










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> <msubsup> <mi> C </mi> <mi> &#957; </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mo> &#62305; </mo> <mtable> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8744; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#955; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> &#8805; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8744; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#955; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> &#8804; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi> &#8734; </mi> <mo> ~ </mo> </mover> </mtd> <mtd> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#955; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> &lt; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#955; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mtd> <mtd> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> &#957; </mi> </msup> </mrow> <mi> &#960; </mi> </mfrac> </mrow> </mtd> <mtd> <mrow> <mrow> <mi> &#955; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#955; </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> &#957; </mi> </msup> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#955; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#955; </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox[&quot;True&quot;, &quot;PiecewiseDefault&quot;, Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> &#957; </ci> </apply> <ci> &#955; </ci> </apply> <ci> z </ci> </apply> <piecewise> <piece> <cn type='integer'> 0 </cn> <apply> <or /> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <and /> <apply> <in /> <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> <ci> &#8469; </ci> </apply> <apply> <or /> <apply> <and /> <apply> <in /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <ci> &#8469; </ci> </apply> <apply> <geq /> <apply> <plus /> <ci> &#955; </ci> <ci> &#957; </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <and /> <apply> <in /> <ci> &#957; </ci> <ci> &#8469; </ci> </apply> <apply> <leq /> <apply> <plus /> <ci> &#955; </ci> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </piece> <piece> <apply> <ci> OverTilde </ci> <infinity /> </apply> <apply> <and /> <apply> <in /> <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> <ci> &#8469; </ci> </apply> <apply> <in /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <ci> &#8469; </ci> </apply> <apply> <lt /> <apply> <plus /> <ci> &#955; </ci> <ci> &#957; </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </piece> <piece> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <apply> <power /> <ci> z </ci> <ci> &#957; </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> &#955; </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> &#955; </ci> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </piece> <piece> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <ln /> <ci> z </ci> </apply> <apply> <power /> <ci> z </ci> <ci> &#957; </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <plus /> <ci> &#955; </ci> <ci> &#957; </ci> </apply> <cn type='integer'> 0 </cn> </apply> </piece> <piece> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <ci> &#957; </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <ci> Gamma </ci> <ci> &#955; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <lt /> <apply> <real /> <apply> <plus /> <ci> &#955; </ci> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </piece> <otherwise> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> &#957; </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> &#955; </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <ci> &#957; </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <ci> Gamma </ci> <ci> &#955; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </otherwise> </piecewise> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List["0", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "\[Nu]"]], ">", "0"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]"]], "\[GreaterEqual]", FractionBox["1", "2"]]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]"]], "\[LessEqual]", RowBox[List["-", FractionBox["1", "2"]]]]]]], ")"]]]], ")"]]]], ")"]]]]], List["ComplexInfinity", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Lambda]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]"]], "<", FractionBox["1", "2"]]]]]], List[FractionBox[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]], RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Lambda]", "+", "\[Nu]"]], "]"]], ">", "0"]]], List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "+", "1"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["Log", "[", "z", "]"]], " ", SuperscriptBox["z", "\[Nu]"]]], "\[Pi]"]]], RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]"]], "\[Equal]", "0"]]], List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], RowBox[List[RowBox[List["Re", "[", RowBox[List["\[Lambda]", "+", "\[Nu]"]], "]"]], "<", "0"]]], List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Nu]"]], "]"]], " ", SuperscriptBox["z", "\[Nu]"]]], RowBox[List[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.