Rational functions approximate a complex function far more efficiently than polynomials when the function has poles, essential singularities, or branch points, because their own poles can migrate to surround those singularities. Convergence is superexponential for entire and meromorphic functions and root-exponential for branch points on the approximation boundary, while polynomials are limited by the nearest singularity. The Walsh-Gonchar potential theory explains these rates by treating the approximant's poles and interpolation points as opposite electric charges whose equilibrium controls the error, and the AAA algorithm computes such near-best rational approximants automatically from sample data.
An expository review by Trefethen of how modern algorithms transformed rational approximation in the complex plane. A function f on a compact, simply connected set K is approximated by rational funct…