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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==-1 > For the function itself > Generic formulas for main term





http://functions.wolfram.com/07.14.06.0067.01









  


  










Input Form





GegenbauerC[\[Nu], \[Lambda], z] \[Proportional] Piecewise[{{0, Element[-\[Nu], Integers] && -\[Nu] > 0}, {ComplexInfinity, Element[-\[Nu] - 2 \[Lambda], Integers] && -\[Nu] - 2 \[Lambda] >= 0}, {-((2^(1/2 - \[Lambda]) Sin[Pi \[Nu]] Gamma[-(1/2) + \[Lambda]] (1 + z)^(1/2 - \[Lambda]))/(Sqrt[Pi] Gamma[\[Lambda]])) + ((Sin[\[Nu] Pi] Gamma[\[Nu] + 2 \[Lambda]])/(Pi Gamma[\[Nu] + 1] Gamma[2 \[Lambda]])) (EulerGamma + Log[(1 + z)/2] - PolyGamma[1/2 + \[Lambda]] + PolyGamma[-\[Nu]] + PolyGamma[2 \[Lambda] + \[Nu]]), Element[\[Lambda] - 3/2, Integers] && \[Lambda] - 3/2 >= 0 && !(Element[\[Nu], Integers] && \[Nu] >= 0)}, {(Sin[\[Nu] Pi]/Pi) (Log[(z + 1)/2] + 2 EulerGamma + PolyGamma[-\[Nu]] + PolyGamma[1 + \[Nu]]), \[Lambda] == 1/2 && !(Element[\[Nu], Integers] && \[Nu] >= 0)}, {(2^(1 - 2 \[Lambda]) Cos[Pi (\[Nu] + \[Lambda])] (-(1/2) - \[Lambda])! Gamma[\[Nu] + 2 \[Lambda]])/(Sqrt[Pi] Gamma[\[Lambda]] Gamma[\[Nu] + 1]) + (-((2^(1/2 - \[Lambda]) Cos[Pi (\[Nu] + \[Lambda])])/ ((1/2 - \[Lambda])! Sqrt[Pi] Gamma[\[Lambda]]))) (z + 1)^(1/2 - \[Lambda]) (EulerGamma + Log[(1 + z)/2] - PolyGamma[3/2 - \[Lambda]] + PolyGamma[1/2 - \[Lambda] - \[Nu]] + PolyGamma[1/2 + \[Lambda] + \[Nu]]), Element[-(1/2) - \[Lambda], Integers] && -(1/2) - \[Lambda] >= 0 && !(Element[\[Nu], Integers] && \[Nu] >= 0)}}, -((2^(1/2 - \[Lambda]) Sin[Pi \[Nu]] Gamma[-(1/2) + \[Lambda]] (1 + z)^(1/2 - \[Lambda]))/(Sqrt[Pi] Gamma[\[Lambda]])) + (2^(1 - 2 \[Lambda]) Cos[Pi (\[Lambda] + \[Nu])] Gamma[1/2 - \[Lambda]] Gamma[2 \[Lambda] + \[Nu]])/(Sqrt[Pi] Gamma[\[Lambda]] Gamma[1 + \[Nu]])] /; (z -> -1)










Standard Form





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MathML Form







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</mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> &#955; </mi> </mrow> </msup> </mrow> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <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> <mrow> <mi> &#955; </mi> <mo> - </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#957; </mi> <mo> &#8713; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mi> &#960; 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Date Added to functions.wolfram.com (modification date)





2007-05-02





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