  The traditional view in numerical conformal mapping is that once the boundary
correspondence function has been found, the map and its inverse can be
evaluated by contour integrals. We propose that it is much simpler, and 10-1000
times faster, to represent the maps by rational functions computed by the AAA
algorithm. To justify this claim, first we prove a theorem establishing
root-exponential convergence of rational approximations near corners in a
conformal map, generalizing a result of D. J. Newman in 1964. This leads to the
new algorithm for approximating conformal maps of polygons. Then we turn to
smooth domains and prove a sequence of four theorems establishing that in any
conformal map of the unit circle onto a region with a long and slender part,
there must be a singularity or loss of univalence exponentially close to the
boundary, and polynomial approximations cannot be accurate unless of
exponentially high degree. This motivates the application of the new algorithm
to smooth domains, where it is again found to be highly effective.
