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

Autorentext

Friedrich Haslinger, University of Vienna, Austria.

Zusammenfassung

"This is a succinct and fairly demanding book, but it is one that repays serious effort. The first half alone, a well-written and elegant exposition of the single variable theory for mathematically mature readers, is worth the price of admission. In addition, because there are not a lot of recent textbooks that cover the multi-variable theory [...], the second half is novel and adds to the value of this book. Professional analysts, or those who aspire to become professional analysts, will want to own this book." MAA Reviews

"The book is clearly written, in a pleasant style and an elegant layout. The first part can be used for graduate courses in complex analysis, while the more advanced second part is adequate for postgraduate courses, as an introductive text on applications of operator theory on Hilbert spaces of holomorphic functions to partial differential equations in complex variables."
S. Cobzas in: Stud. Univ. Babes-Bolyai Math. 63/2 (2018), 285-286