Figure 4.29.4: Principal values of $\mathrm{arctanh}x$ and $\mathrm{arccoth}x$.
($\mathrm{arctanh}x$ is complex when $$ or $x>1$, and
$\mathrm{arccoth}x$ is complex when $$.)
Magnify

The conformal mapping $w=\mathrm{sinh}z$ is obtainable from Figure
4.15.7 by rotating both the $w$-plane and the $z$-plane
through an angle $\frac{1}{2}\pi $, compare (4.28.8).

The surfaces for the complex hyperbolic and inverse hyperbolic functions are
similar to the surfaces depicted in §4.15(iii) for the
trigonometric and inverse trigonometric functions. They can be visualized with
the aid of equations (4.28.8)–(4.28.13).