The hyperbolic sine is an always increasing function, therefore it is bijective and invertible over all ℜ. If we denote by arcsinh its inverse function, we can write
From the second one of the (3.9), remembering that ey>0, we have
and finally
The hyperbolic cosine isn't monotonic, so it isn't invertible over all ℜ; to make it invertible we must restrict its range to non negative arguments. Only in this case we can have its inverse function, denoted by arccosh.
From the first one of the (3.9), excluding the negative y,
and finally
The hyperbolic tangent is an always increasing function, therefore it is bijective and invertible over all ℜ. If we denote by arctanh its inverse function, we can write
From the third one of the (3.9), remembering that e2y>0, we have
