As part of a larger (non-maths) research project, we're looking for a conformal map from a 'cutoff cylinder' created by cutting away a part of a disk on each side and gluing along those sides, onto the annulus. I've attempted to draw this clearly in the picture below. We know how to go from an 'ordinary' cylinder, that's pretty straight-forward, but we've been having some issues with the cutoff case. We're hoping this might be a relatively easy problem for someone who's better at conformal maps than we are. Any ideas would be appreciated!
Thank you!
Such a map exists by the Rieman mapping theorem. Indeed, the unit square and the flatten disc both satisfy the conditions of the theorem and then, by transitivity, you can map the disc to the annulus biholomorphically. Unfortunately, it is not usually possible to express this function explicitly. For further reading see, for example, proof of the RMT.