Complex Analysis

An Invitation

This volume is an enlarged edition of a classic textbook on complex analysis. In addition to the classical material of the first edition it provides a concise and accessible treatment of Loewner theory, both in the disc and in the half-plane. Some of the new material has been described in research papers only or appears here for the first time. Each chapter ends with exercises.

Contents:

• Power Series
• Holomorphic and Analytic Functions
• Exponential Function, Logarithm and Winding Number
• Basic Theory of Holomorphic Functions
• Global Theory
• Isolated Singularities
• The Picard Theorems
• Geometric Aspects and the Riemann Mapping Theorem
• Meromorphic Functions and Runge's Theorems
• Representations of Meromorphic Functions
• The Prime Number Theorem
• Harmonic Functions
• Subharmonic Functions
• Various Applications
• Jordan Regions
• The Dirichlet Problem and Green's Functions
• Extending Riemann Maps to the Boundary
• Carathéodory Convergence
• Capacities
• Loewner Theory

Readership: Advanced undergraduates and graduate students of mathematics including mathematicians interested in Loewner theory.
Key Features:

• It contains an accessible introduction to the results and techniques of complex function theory with recent simplifications
• It discusses compact hulls, half-plane hulls and their capacities. Examples are computed in detail
• It contains recent results in Loewner theory and proofs of them, not just in the disc, but also in the half-plane