In this textbook, a concise approach to complex analysis of one and several variables is presented. After an introduction of Cauchy‘s integral theorem general versions of Runge‘s approximation theorem and Mittag-Leffler‘s theorem are discussed. The fi rst part ends with an analytic characterization of simply connected domains. The second part is concerned with functional analytic methods: Fréchet and Hilbert spaces of holomorphic functions, the Bergman kernel, and unbounded operators on Hilbert spaces to tackle the theory of several variables, in particular the inhomogeneous Cauchy-Riemann equations and the d-bar Neumann operator.

Contents

Complex numbers and functions

Cauchy’s Theorem and Cauchy’s formula

Analytic continuation

Construction and approximation of holomorphic functions

Harmonic functions

Several complex variables

Bergman spaces

The canonical solution operator to

Nuclear Fréchet spaces of holomorphic functions

The -complex

The twisted -complex and Schrödinger operators