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

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2007-05-02