08-17-2026, 03:41 PM
Functions of One Complex Variable
Author: John B. Conway
First published: 1973
Publisher: Springer-Verlag, New York
John B. Conway’s Functions of One Complex Variable is a rigorous and influential introduction to complex analysis, designed primarily as a first serious course for mathematically mature students. Conway assumes relatively little beyond calculus and some knowledge of partial derivatives, but the treatment quickly develops the level of rigor expected in graduate mathematics. An important feature of the book is its perspective: complex analysis is presented not merely as a collection of computational techniques but as an entry point into broader mathematical ideas involving analysis, topology and geometry.
The book begins with the complex number system and the topology of $\mathbb C$, then develops analytic functions and complex integration. From there Conway treats the central results of classical complex analysis, including singularities, the maximum modulus theorem, convergence and compactness of families of analytic functions, and Runge's theorem. The later chapters move toward deeper topics such as analytic continuation, Riemann surfaces, harmonic functions, entire functions, and the range of analytic functions. This progression makes the book substantially more than a computational introduction: the reader gradually sees how local properties of holomorphic functions produce remarkably strong global consequences.
A major strength is the balance between rigor and breadth. Proofs and definitions are treated carefully, while the selection of material is broad enough for approximately a full-year course. The style is theorem-oriented and requires active mathematical reading, so it is considerably more demanding than elementary complex-variable texts. For a student interested in analysis or preparing for graduate mathematics, however, that difficulty is precisely part of its value: Conway develops both complex analysis itself and the habits of rigorous reasoning needed for more advanced subjects. MathSciNet's assessment similarly emphasizes the book's careful mathematical and pedagogical treatment and its suitability for classroom study or self-study.
Key takeaways
Goodreads — Functions of One Complex Variable
Author: John B. Conway
First published: 1973
Publisher: Springer-Verlag, New York
John B. Conway’s Functions of One Complex Variable is a rigorous and influential introduction to complex analysis, designed primarily as a first serious course for mathematically mature students. Conway assumes relatively little beyond calculus and some knowledge of partial derivatives, but the treatment quickly develops the level of rigor expected in graduate mathematics. An important feature of the book is its perspective: complex analysis is presented not merely as a collection of computational techniques but as an entry point into broader mathematical ideas involving analysis, topology and geometry.
The book begins with the complex number system and the topology of $\mathbb C$, then develops analytic functions and complex integration. From there Conway treats the central results of classical complex analysis, including singularities, the maximum modulus theorem, convergence and compactness of families of analytic functions, and Runge's theorem. The later chapters move toward deeper topics such as analytic continuation, Riemann surfaces, harmonic functions, entire functions, and the range of analytic functions. This progression makes the book substantially more than a computational introduction: the reader gradually sees how local properties of holomorphic functions produce remarkably strong global consequences.
A major strength is the balance between rigor and breadth. Proofs and definitions are treated carefully, while the selection of material is broad enough for approximately a full-year course. The style is theorem-oriented and requires active mathematical reading, so it is considerably more demanding than elementary complex-variable texts. For a student interested in analysis or preparing for graduate mathematics, however, that difficulty is precisely part of its value: Conway develops both complex analysis itself and the habits of rigorous reasoning needed for more advanced subjects. MathSciNet's assessment similarly emphasizes the book's careful mathematical and pedagogical treatment and its suitability for classroom study or self-study.
Key takeaways
- Rigorous foundation: Builds complex analysis systematically from $\mathbb C$ and topology through the major classical theorems.
- Broad mathematical viewpoint: Connects complex analysis naturally with topology, harmonic analysis and Riemann surfaces.
- Substantial coverage: Goes beyond Cauchy's theorem and residues to Runge approximation, analytic continuation, entire functions and related advanced topics.
- Best suited to serious study: An excellent choice for advanced undergraduates, graduate students, or mathematically mature self-learners who want a proof-oriented treatment rather than primarily computational techniques.
Goodreads — Functions of One Complex Variable
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