## 6. Inverse hyperbolic functions

### arcsinh

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 *e*^{y}>0, we have

and finally

### arccosh

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

### arctanh

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 *e*^{2y}>0, we have

and finally