Theory of Functions of a Complex Variable: Authorized Translation from the Fourth German Edition (Classic Reprint)

Explore the theory of functions of a complex variable, presented as an accessible, unified introduction

This edition surveys the central ideas in complex analysis, weaving together geometric intuition and rigorous proofs. It blends classical approaches and modern developments to help beginners grasp how complex numbers behave under analytic methods, with careful attention to foundational concepts and key theorems.

The book opens by building the algebra and geometry of complex numbers, then moves through rational and again to many-valued functions. It emphasizes conformal representations, the role of the RIEMANN surface, and the historic connections between RIEMANN, WEIERSTRASS, and other pioneers. The exposition introduces the exponential and trigonometric functions for complex arguments, including their addition theorems and fundamental relations, before turning to singularities, Laurent series, and the general theory of analytic continuation. A clear pathway is provided for readers who want both theoretical understanding and practical problem-solving tools.

• Foundational concepts: complex numbers, functions, and geometric representation
• Core theorems on regular and single-valued analytic functions
• Techniques in conformal mapping, rational and algebraic functions
• Expanded treatment of exponential, trigonometric, and multivalued functions

Ideal for readers of math and analysis who seek a compact, rigorous introduction to the theory of complex variables.