Sharp constants for rational approximations of analytic functions

Theorems describing the sharp constants for the approximation of a general class of analytic functions by rational functions are proved. Magnus's conjecture on the sharp constant for the approximation of $e^{-z}$ on $[0,\infty]$ is established as a consequence. For the proof of the theorems new formulae expressing the strong asymptotics of polynomials orthogonal with respect to a varying complex weight are obtained.