That is, if R1/r1 does not equal R2/r2 then there is no conformal map between the two regions.
So suppose you have to rings, both with inner radius 1.
One has outer radius 2 and the other has outer radius 3.
Then these rings are not conformally equivalent.
They look a lot more like each other than do Mickey Mouse and Batman, but the latter are conformally equivalent and the former are not.
Almost any two regions without holes are conformally equivalent, but it’s much harder for two regions with holes to be conformally equivalent.
Related postsJacobi functionsApplied complex analysis[1] Conformal maps also preserve orientation.
So if you move clockwise from the first curve to the second in the domain, you move clockwise from the image of the first to the image of the second by the same angle.
A function of a complex variable is conformal if and only if it is holomorphic, i.
e.
analytic on the interior of its domain, with non-zero derivative.
Some authors leave out the orientation preserving part of the definition; under their weaker definition a conformal map could also be the conjugate of a holomophic function.
[2] A rigorous way to say that a region has no holes is to say its fundamental group is trivial.
Any loop inside the region is homotopic to a point, i.
e.
you can continuously shrink it to a point without leaving the region.
[3] Liouville’s theorem says an analytic function that is bounded on the entire plane must be constant.
That’s why the Riemann mapping theorem says a region must be a proper subset of the plane, not the whole plane.
.