I've been looking into conformal maps recently to try and solve a problem I'm looking at. I'd like to flatten a coordinate system from polar onto cartesian, and transform all of the geometry from one coordinate system to the other.
Specifically, I'd like to take a point at $z=i$ on the unit disk, and map that point onto the origin $z' = 0$ of a cartesian coordinate system. I'd also like to take points $z = -i \pm \epsilon$ where epsilon is a positive infinitesimal and map that to a point $z' = \pm \pi$. Any point $z > |1|$, I'd like to map to a cartesian point like $z = (r,\theta)$ where $r > 1$ maps to $z' = (x,iy)$ as $r = i(y+1)$ and $x = \theta$.
I've figured out how to map the disk onto the upper half plane, but this stretches the lengths $-2i \pm \epsilon$ out to $\pm \infty$, and I'd prefer to have the map confined to just a strip.
Is there a map that does this? Computationally, it's easy enough to set y = r-1, and x = theta, but I'm hoping to map functions and data as well. This is kind of a new field of math for me, so please forgive me if I'm asking rather naïve questions.