Mittag-Leffler Theorem and Series Representation of Meromorphic Functions

Abstract

The goal of this thesis is to present a complete proof of Mittag-Leffler theorem, and to present some applications. We first give a simpler proof, valid for open disks and the whole complex plane. This proof uses in a crucial way the approximation of holomorphic function by means of polynomials on such domains, by means of their Taylor series. The proof of Mittag-Leffler theorem for an arbitrary domain G ⊂ ℂ is much more involved and requires the development of a more general theory of approximation of holomorphic function on complex domains. We present this theory and prove Runge’s theorem, a fundamental result in complex analysis providing conditions to have the required approximations. This allows us to complete the proof of the general version of Mittag-Leffler theorem. We complete this thesis by presenting some applications of the above results.