Mapping in complex analysis has not been very easy for me unfortunately. I am having difficult trying to find the mapping between the z and w plane. I attempted to simply write that
$w=\cosh^{-1}{z}$
and applying
$\cosh^{-1}{z}=\log(z+(z^2-1)^{1/2})$
and trying to plug in $z=x+iy$ to get a relation between $z$ and $w$ such that w:
$w=u(x,y)+iv(x,y)$
Is the way to go about it. This is how I have been handling mapping problems but there doesn't seem to be a clear cut way of doing the simplification in this problem.