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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 on branch cuts > For the function itself





http://functions.wolfram.com/07.14.06.0042.01









  


  










Input Form





GegenbauerC[\[Nu], \[Lambda], z] \[Proportional] ((2^(1 - 2 \[Lambda]) Cos[Pi (\[Lambda] + \[Nu])] Sin[Pi \[Nu]])/ (Pi^(3/2) Gamma[\[Lambda]])) ((-I) Pi^(3/2) 4^\[Lambda] E^(I Pi (1/2 - \[Lambda]) Floor[Arg[z - x]/(2 Pi)]) Csc[Pi \[Nu]] Floor[Arg[z - x]/(2 Pi)] Gamma[\[Lambda]] GegenbauerC[\[Nu], \[Lambda], -x] - Exp[Pi I (1 - 2 \[Lambda]) Floor[Arg[z - x]/(2 Pi)]] MeijerG[{{1 + \[Nu], 1 - 2 \[Lambda] - \[Nu]}, {}}, {{0, 1/2 - \[Lambda]}, {}}, (1/2) (1 + x)] + (1/2) (2 Pi I E^(I Pi (1/2 - \[Lambda]) Floor[Arg[z - x]/(2 Pi)]) Floor[Arg[z - x]/(2 Pi)] Gamma[1 - \[Nu]] Gamma[1 + 2 \[Lambda] + \[Nu]] Hypergeometric2F1Regularized[1 - \[Nu], 1 + 2 \[Lambda] + \[Nu], 3/2 + \[Lambda], (1/2) (1 + x)] + Exp[Pi I (1 - 2 \[Lambda]) Floor[Arg[z - x]/(2 Pi)]] MeijerG[{{\[Nu], -2 \[Lambda] - \[Nu]}, {}}, {{0, -(1/2) - \[Lambda]}, {}}, (1/2) (1 + x)]) (z - x) + (1/8) (2 Pi I E^(I Pi (1/2 - \[Lambda]) Floor[Arg[z - x]/(2 Pi)]) Floor[Arg[z - x]/(2 Pi)] Gamma[2 - \[Nu]] Gamma[2 + 2 \[Lambda] + \[Nu]] Hypergeometric2F1Regularized[2 - \[Nu], 2 + 2 \[Lambda] + \[Nu], 5/2 + \[Lambda], (1/2) (1 + x)] - Exp[Pi I (1 - 2 \[Lambda]) Floor[Arg[z - x]/(2 Pi)]] MeijerG[{{-1 + \[Nu], -1 - 2 \[Lambda] - \[Nu]}, {}}, {{0, -(3/2) - \[Lambda]}, {}}, (1/2) (1 + x)]) (z - x)^2 + \[Ellipsis]) /; (z -> x) && Element[x, Reals] && x < -1










Standard Form





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MathML Form







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</ci> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <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> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> &#955; </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <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> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </list> <list /> </list> <list> <list> <cn type='integer'> 0 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </list> <list /> </list> <apply> <times /> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <ci> x </ci> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02