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 GegenbauerC

 http://functions.wolfram.com/07.14.06.0080.01

 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]]) + ((2^(1 - 2 \[Lambda] - \[Nu]) Cos[Pi (\[Lambda] + \[Nu])] Gamma[2 \[Lambda] + \[Nu]] Sin[Pi \[Nu]] z^(-2 \[Lambda] - \[Nu]))/(Pi Gamma[\[Lambda]] Gamma[1 + \[Lambda] + \[Nu]])) (EulerGamma + Log[2] - Log[z] + PolyGamma[1/2 - \[Lambda] - \[Nu]] + PolyGamma[2 \[Lambda] + \[Nu]] - PolyGamma[1 + 2 \[Lambda] + 2 \[Nu]]), Element[2 (\[Lambda] + \[Nu]), Integers] && 2 (\[Lambda] + \[Nu]) > 0 && NotElement[1/2 + \[Lambda] + \[Nu], Integers]}, {((2^(1 + \[Nu]) z^\[Nu] Sin[Pi \[Nu]])/Pi) (EulerGamma - Log[2] - Log[z] + PolyGamma[-\[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]])) - ((2^(1 + \[Nu]) Cos[Pi (\[Lambda] + \[Nu])] z^\[Nu])/(Gamma[\[Lambda]] Gamma[1 - \[Lambda] - \[Nu]] Gamma[1 + \[Nu]])) (EulerGamma + Log[2] - Log[z] - PolyGamma[1 - 2 \[Lambda] - 2 \[Nu]] + PolyGamma[-\[Nu]] + PolyGamma[1/2 + \[Lambda] + \[Nu]]), Element[-2 (\[Lambda] + \[Nu]), Integers] && -2 (\[Lambda] + \[Nu]) > 0 && NotElement[1/2 - \[Lambda] - \[Nu], Integers]}, {(2^\[Nu] Gamma[\[Lambda] + \[Nu]] z^\[Nu])/(Gamma[\[Lambda]] Gamma[1 + \[Nu]]), Element[-(1/2) - \[Lambda], Integers] && -(1/2) - \[Lambda] >= 0 && Element[\[Nu], Integers] && \[Nu] >= 0 && \[Lambda] + \[Nu] > -(1/2)}}, -((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

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

 Rule Form

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

 2007-05-02