asinh(x)

Now consider the hyperbolic sine in terms of the exponential. Solving for this gives

$\begin{array}{c}sinhw=\frac{e^w-e^{-w}}{2}=z\\ e^{2w}-2z{e}^w-1=0\\ e^w=z\pm \sqrt{z^2+1}\\ w={sinh}^{-1}z=ln(z\pm \sqrt{z^2+1})\end{array}$

The sign from the square root is a new twist that will apply here and in subsequent functions. This branching behavior exists in addition to that of the logarithm and must be included for a full picture of the complex structure. This will be made clear in the visualizations.

Applying the behavior of the logarithm, the inverse hyperbolic sine on an arbitrary branch is

${sinh}^{-1}z=ln(z\pm \sqrt{z^2+1})+2\pi ni$

The individual branches look like this:

One half of each branch comes from using the plus sign on the square root, the other half from the minus sign. The lines where clearly discordant colors meet are where transitions to higher or lower branches occur.

The real part of this function retains the same numerical value between branches, while the imaginary part moves up and down in value. Visualize the imaginary part of several branches simultaneously: