I have to find the transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ that goes from the 2D annulus to a square, and I am unsure if my procedure is right.
More precisely, the problem is as follows:
Find the transformation from the domain $A$ to $B$, where $A$ is described according to $(\xi, \eta) \in A = [-1,1]^2$ and $(r,\theta)\in B \subseteq \mathbb{R}^2$ where $r \in [r_0, r_0 + \Delta r]$ and $\theta \in [-\frac{\Delta \theta}{2}, \frac{\Delta \theta}{2}]$
So I found from http://www-users.math.umn.edu/~olver/ln_/cml.pdf (page 29) that this is equivalent to the conformal mapping
$$(x, y) = (e^\xi\cos \eta, e^\xi \sin \eta)$$
But how do I constructively come up with this conclusion?