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

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2007-05-02