Is there a conformal mapping that transforms regular polygons (e.g. triangle and square) to circle?
I would hazard a guess and say no by definition of a conformal mapping angles are preserved but in mapping the polygon to the circle you lost the angles right?
Surprisingly, yes, there is.
The problem with this theorem, though, is that the mappings might not be that easy to get explicitly.
As homeomorphic wrote, the Riemann Mapping Theorem solves this problem.
A couple of points for clarification
The theorem applies to non-empty simply connected open domains in the complex plane other than the entire plane. The interior of a polygon and a circle are both simply connected. Thus there is a conformal mapping from the interior of the polygon onto the interior of the circle. The bounding polygon is mapped onto the bounding circle but the map is obviously not conformal on these edges and can never be ( as you suspected).
The boundary of a simply connected domain can be complicated and need not be piece wise smooth. Nevertheless its interior can be mapped conformally onto the interior of a circle.
Thank you all!
The Schwarz-Christoffel transformation will map a polygon to the upper half plane, and a Mobius transformation can map a half-plane to the unit circle.
Separate names with a comma.