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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





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MathML Form







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</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





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





2007-05-02