Complex Analysis: A Functional Analytic Approach

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 first 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

A modern approach to complex analysis of one and multiple variables.

Covers multiple variables using methods of functional analysis.

Well suited for introductory and advanced courses on complex analysis.