Conformal mapping may be the best-known topic in complex analysis. Any simply connected nonempty domain $\Omega$ in the complex plane ${ { \mathbb{ C}}}$ (assuming $\Omega\ne { { \mathbb{ C}}}$) can be mapped bijectively to the unit disk by an analytic function with nonvanishing derivative, as in Figure 1. If $\Omega$ is doubly-connected, it can be mapped to a circular annulus $1<|z|<R$ for some $R$, called the conformal modulus, which is uniquely determined by $\Omega$, as in Figure 2. If $\Omega$ has connectivity higher than $2$, it can be mapped onto various canonical domains such as a disk with exclusions in the form of slits or smaller disks, as in Figure 3.
Categories: math.CV