ChebyshevT[\[Nu], z] == (-((2^(-2 - \[Nu]) (-z^2)^(\[Nu]/2))/(Pi Gamma[-\[Nu]]))) (Cos[(Pi \[Nu])/2] + (z Sin[(Pi \[Nu])/2])/Sqrt[-z^2]) (Sum[Residue[((Gamma[s] Gamma[1/2 + s] Gamma[-s - \[Nu]/2])/ (1 - 1/z^2)^s) Gamma[-s + (1 - \[Nu])/2], {s, j + (1 - \[Nu])/2}], {j, 0, Infinity}] + Sum[Residue[((Gamma[s] Gamma[1/2 + s] Gamma[-s + (1 - \[Nu])/2])/ (1 - 1/z^2)^s) Gamma[-s - \[Nu]/2], {s, j - \[Nu]/2}], {j, 0, Infinity}]) - (2^(-2 + \[Nu])/((-z^2)^(\[Nu]/2) (Pi Gamma[\[Nu]]))) (Cos[(Pi \[Nu])/2] - (z Sin[(Pi \[Nu])/2])/ Sqrt[-z^2]) (Sum[Residue[((Gamma[s] Gamma[1/2 + s] Gamma[-s + (1 + \[Nu])/2])/ (1 - 1/z^2)^s) Gamma[-s + \[Nu]/2], {s, j + \[Nu]/2}], {j, 0, Infinity}] + Sum[Residue[((Gamma[s] Gamma[1/2 + s] Gamma[-s + \[Nu]/2])/ (1 - 1/z^2)^s) Gamma[-s + (1 + \[Nu])/2], {s, j + (1 + \[Nu])/2}], {j, 0, Infinity}]) /; Abs[1 - 1/z^2] > 1 && !Element[2 \[Nu], Integers]

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2007-05-02