EllipticPi[n, z, m] \[Proportional] (-1)^Round[Re[z]/Pi] Csc[z] ((-((Pi Sqrt[(-n) Sin[z]^2])/(2 n))) (1 + (1 + m)/(2 n) + (3 + 2 m + 3 m^2)/(8 n^2) + \[Ellipsis]) + ((1/n) AppellF1[-(1/2), 1/2, 1/2, 1/2, m Sin[z]^2, Sin[z]^2] + (1/(3 n^2 z^2)) AppellF1[-(3/2), 1/2, 1/2, -(1/2), m Sin[z]^2, Sin[z]^2] + (1/(5 n^3 z^4)) AppellF1[-(5/2), 1/2, 1/2, -(3/2), m Sin[z]^2, Sin[z]^2] + \[Ellipsis])) + Round[Re[z]/Pi] ((Pi/Sqrt[-n]) (1 + (1 + m)/(2 n) + (3 + 2 m + 3 m^2)/(8 n^2) + \[Ellipsis]) + (1/(2 n)) (4 (EllipticE[m] - EllipticK[m]) - (-8 (1 + m) EllipticE[m] + 4 (2 + m) EllipticK[m])/(3 n) + (8 ((8 + 7 m + 8 m^2) EllipticE[m] - (8 + 3 m + 4 m^2) EllipticK[m]))/ (15 n^2) + \[Ellipsis])) /; (Abs[n] -> Infinity)

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2007-05-02