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 GegenbauerC

 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

 C ν λ ( z ) 0 - ν + ~ - 2 λ - ν sin ( ν π ) Γ ( 2 λ + ν ) π Γ ( ν + 1 ) Γ ( 2 λ ) ( log ( z + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( λ + 1 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( - ν ) + ψ TagBox["\[Psi]", PolyGamma] ( 2 λ + ν ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) - 2 1 2 - λ sin ( π ν ) Γ ( λ - 1 2 ) ( z + 1 ) 1 2 - λ π Γ ( λ ) λ - 3 2 ν sin ( ν π ) π ( log ( z + 1 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( - ν ) + ψ TagBox["\[Psi]", PolyGamma] ( ν + 1 ) + 2 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) λ 1 2 ν 2 1 - 2 λ cos ( π ( λ + ν ) ) ( - λ - 1 2 ) ! Γ ( 2 λ + ν ) π Γ ( λ ) Γ ( ν + 1 ) - 2 1 2 - λ cos ( π ( λ + ν ) ) ( z + 1 ) 1 2 - λ ( 1 2 - λ ) ! π Γ ( λ ) ( log ( z + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( 3 2 - λ ) + ψ TagBox["\[Psi]", PolyGamma] ( - λ - ν + 1 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( λ + ν + 1 2 ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) - λ - 1 2 ν 2 1 - 2 λ cos ( π ( λ + ν ) ) Γ ( 1 2 - λ ) Γ ( 2 λ + ν ) π Γ ( λ ) Γ ( ν + 1 ) - 2 1 2 - λ sin ( π ν ) Γ ( λ - 1 2 ) ( z + 1 ) 1 2 - λ π Γ ( λ ) True TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] Proportional Subscript C ν λ z 0 -1 ν SuperPlus OverTilde -2 λ -1 ν ν Gamma 2 λ ν Gamma ν 1 Gamma 2 λ -1 z 1 2 -1 -1 PolyGamma λ 1 2 PolyGamma -1 ν PolyGamma 2 λ ν -1 2 1 2 -1 λ ν Gamma λ -1 1 2 z 1 1 2 -1 λ 1 2 Gamma λ -1 λ -1 3 2 ν ν -1 z 1 2 -1 PolyGamma -1 ν PolyGamma ν 1 2 λ 1 2 ν 2 1 -1 2 λ λ ν -1 λ -1 1 2 Gamma 2 λ ν 1 2 Gamma λ Gamma ν 1 -1 -1 2 1 2 -1 λ λ ν z 1 1 2 -1 λ 1 2 -1 λ 1 2 Gamma λ -1 z 1 2 -1 -1 PolyGamma 3 2 -1 λ PolyGamma -1 λ -1 ν 1 2 PolyGamma λ ν 1 2 -1 λ -1 1 2 ν 2 1 -1 2 λ λ ν Gamma 1 2 -1 λ Gamma 2 λ ν 1 2 Gamma λ Gamma ν 1 -1 -1 2 1 2 -1 λ ν Gamma λ -1 1 2 z 1 1 2 -1 λ 1 2 Gamma λ -1 [/itex]

 Rule Form

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

 2007-05-02

© 1998-2013 Wolfram Research, Inc.