ArcTanh[x] + ArcCosh[y] == ArcTanh[(Sqrt[1 - x^2] Sqrt[1 - (I x y + I Sqrt[y - 1] Sqrt[y + 1])^2/ (1 - x^2)])/ ((-1)^Floor[1/2 - Arg[(Sqrt[y - 1] Sqrt[y + 1] x + y)/Sqrt[1 - x^2]]/Pi] (x y + Sqrt[y - 1] Sqrt[y + 1]))] - (1/2) I (-1)^(Floor[1/2 - Arg[(Sqrt[y - 1] Sqrt[y + 1] x + y)/Sqrt[1 - x^2]]/Pi] + Floor[1/2 - Arg[1 - x^2]/(2 Pi) + Arg[I x y + I Sqrt[-1 + y] Sqrt[1 + y]]/Pi]) Pi - (1/4) (1 + (-1)^(-Floor[-(Arg[1 - y]/(2 Pi))])) Pi I (2 (1 + (-1)^Floor[1/2 - Arg[(I Sqrt[1 - y^2] x + y)/Sqrt[1 - x^2]]/Pi]) Floor[(Arg[(I - I x)/Sqrt[1 - x^2]] + Arg[I y + Sqrt[1 - y^2]])/ (2 Pi)] + (-1)^Floor[1/2 - Arg[(I Sqrt[1 - y^2] x + y)/Sqrt[1 - x^2]]/ Pi] - 2 (-1 + (-1)^Floor[1/2 - Arg[(I Sqrt[1 - y^2] x + y)/Sqrt[1 - x^2]]/Pi]) Floor[1/2 - (Arg[(I - I x)/Sqrt[1 - x^2]] + Arg[I y + Sqrt[1 - y^2]])/ (2 Pi)] - 1) + (1/4) (1 - (-1)^(-Floor[-(Arg[1 - y]/(2 Pi))])) Pi I (2 (1 + (-1)^Floor[1/2 - Arg[(y - I x Sqrt[1 - y^2])/Sqrt[1 - x^2]]/Pi]) Floor[(Arg[(I x + I)/Sqrt[1 - x^2]] + Arg[I y + Sqrt[1 - y^2]])/ (2 Pi)] + (-1)^Floor[1/2 - Arg[(y - I x Sqrt[1 - y^2])/Sqrt[1 - x^2]]/ Pi] - 2 (-1 + (-1)^Floor[1/2 - Arg[(y - I x Sqrt[1 - y^2])/Sqrt[1 - x^2]]/Pi]) Floor[1/2 - (Arg[(I x + I)/Sqrt[1 - x^2]] + Arg[I y + Sqrt[1 - y^2]])/ (2 Pi)] - 1)

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2007-05-02