Complex Analysis, an Introduction to the Theory of Analytic Functions of One Complex Variable

by Lars Valerian Ahlfors

An Introduction to the Theory of Analytic Functions of one complex variable

Genres: Mathematics, Textbooks, Stem, Science, Reference

247 pages, Hardcover

First published December 1, 1978

About the author

Lars Valerian Ahlfors, William Caspar Graustein Professor of Mathematics, died of pneumonia on Oct. 11 in Pittsfield, Mass., at the age of 89. Ahlfors won the first Fields Medal awarded by the International Mathematics Society in 1936, a quadrennial award considered equivalent, in mathematics, to the Nobel Prize. In 1979, he was awarded the prestigious Wolf Prize in Jerusalem. Ahlfors was best known for his work in complex analysis, a fundamental subject with many applications from number theory to modern physics. His textbook, Complex Analysis, first published in 1953, with new editions in 1966 and 1979, is still considered the leading text in the field. He wrote three other mathematical books and published almost 100 papers. Colleagues and students have described his work as extraordinarily elegant, and his lectures, delivered in thundering basso, as stunningly beautiful. Ahlfors was born in Helsinki, Finland, in 1907. As he often observed, his mother's death during his birth critically influenced his life. His father, a professor of mechanical engineering at the Polytechnic Institute, was attentive but stern. In the brief autobiographical note that introduced his collected papers, published in 1982, he wrote that, "As a child, I was fascinated by mathematics without understanding what it was about, but I was by no means a child prodigy. As a matter of fact, I had no access to any mathematical literature except in the highest grades. . . . The high school curriculum did not include any calculus but I finally managed to learn some on my own, thanks to clandestine visits to my father's engineering library." Ahlfors' introduction to higher mathematics, including complex analysis, came from his mentors at Helsinki University, Ernst Lindelof and Rolf Nevanlinna. At 21, Ahlfors followed Nevanlinna to the Federal Polytechnic Institute in Zurich, where he began working at a research level. There he produced his first major work, a study of asymptotic values of an entire function, based on his own new approach to conformal mapping. Self-effacingly, Ahlfors credited Nevanlinna and another teacher, George Polya, for their "considerable help." They, in turn, insisted that he publish the results solely in his own name. Thereafter, as he expressed it, "I have tried to repay my debt by never accepting to appear as coauthor with a student."

Reviews

[James Swenson · Rating 4 out of 5]

This is the standard graduate textbook in the field. Though readable, it is dense, and suffers from a lack of figures, exercises, and (above all) examples. It is very possible to read this book with pleasure from cover to cover, but end up unable to do anything.

I echo another reviewer: it is best to know calculus perfectly, and to know a fair amount of complex analysis, before using this book for self-study. An instructor selecting this textbook is obligated to supply the examples that every student will need.

[Julia · Rating 5 out of 5]

extraordinary book for beginning through advanced complex analysis, with a focus on the geometric -- but by no means lax. thoughtful, well paced, precise.

[Jovany Agathe · Rating 2 out of 5]

This is an excellent and classic treatment of complex analysis. The treatment is comparatively modern and uses the language of point set topology throughout. However, the author takes pains to develop geometric intuition whenever feasible, without letting the intuitiveness result in a decrease in standards or rigor. This textbook will require a good deal of mathematical maturity, although somewhat more classical in presentation, the book probably requires a bit more mathematical maturity than the modern standard by Stein and Shakarchi. The exercises look challenging, but slightly more staightforward than Stein and Shakarchi. This book should be read for the way a true towering figure in the field of complex analysis thinks about it and transmits it to beginners.

[Ronald Lett · Rating 4 out of 5]

A beautiful exposition of complex analysis. One warning, though: you should have a good understanding of complex algebra and calculus before reading this text, as it is dense.

[Halden Halverson · Rating 5 out of 5]

98/100 - Defining Work - S

Ahlfors' Complex Analysis is a great book, to put it simply. You can see how much Ahlfors respects the subject, and the reader. It's uncompromisingly rigorous and it knows its greatness. Proofs in this book are cogent, and assumptions are clean. Nothing feels hand-wavy here. In terms of conceptual death, it does what few texts manage to do. It manages to balance local arguments with global structure well, and is one of the first analysis texts to do so as cleanly as Ahlfors does. This text is good here.

On the flip side, the book is demanding, but it is great if you're ready to think abstractly. It assumes that you're serious, and then repays that. If you're coming straight from a computational calculus course, or any general course load where you lack experience with proof-based maths, it may be harder to start without supplements. It does also struggle with availability of exercises. One looking for drill-style practice that provides computational comfort may want to supplement with another text.

It did more than be a good analysis book, though. Ahlfors' text defined complex analysis pedagogy for the rest of the 20th century. Complex analysis was originally taught as a collection of computational techniques, such as residue calculus and conformal mappings. It could also be approached by formalizing real-variable methods heavily and extending them, though this is now considered handwavy. Ahlfors' text looked from a geometric-analytic perspective, and did it cogently. He explained why analyticity and geometry can't be separated, which in turn shaped the theory itself.

His manner of explanation also influenced math pedagogy for the latter half of the 20th century. He showed that rigor doesn't need verbosity. In this text, the definitions are generally leaner and with purpose. There isn't much "journal speak" here, and things aren't included "for tradition's sake", while still expecting the reader to think carefully, which led to a change in how complex analysis was taught, explanation-wise.

Overall, this text is great, and I recommend it. The book inspired many later texts by inheriting the idea that complex analysis is centralized on analyticity and conformal invariance. Later texts such as Needham (geometric), Rudin (measure-theoretic), and Stein & Shakarchi (harmonic-analytic) all stemmed from this same idea. I can't give this text anything less than a 98 due to the foothold that this book had on analysis pedagogy, and still has to this day.

[Lucas Chen · Rating 5 out of 5]

read this one after my reu mentor recommended mcmullen's notes (wendy, I'm so so sorry but I am not psychic enough to read a graduate magic-carpet second course having barely ever seen complex in my life). nonetheless probably a good idea to supplement with said notes because exercises are a bit sparse. (as tends to be the case, wendy was in fact completely right 😐)

complex analysis starts as a really convenient subject, but i think ahlfors illustrates its natural beauty right away and while i haven't mastered any part of it, this book made me really excited to learn more about it. (i'm stupid so last year i thought because my program offers a total of one course in pure complex analysis that it must not be that deep.) amazing exposition and clear, motivated tangents — really good jumping off point for me and i can definitely see why this is the classic of the field, unlike for instance baby rudin.

[Ryan · Rating 4 out of 5]

Excellent for shoring up one's understanding of any theorem that might be covered in a Complex Analysis course. Well written and perfectly accessible for any undergraduate.

[Annushka · Rating 2 out of 5]

Uses the word “clearly” with reckless abandon

[Deni Lennon · Rating 4 out of 5]

Used it during my complex analysis class.
Great choice, but you still have to search additional information.

[Tinwerume · Rating 2 out of 5]

I'm only skimming it (too much of it is rehashing things I already know), so I don't think this is a completely fair assessment. Mostly I would like it if theorems and definitions were more clearly marked, but I think if I were reading it in detail that might not annoy me as much.

UPDATE: It's *really* lacking in exercises. It seems okay as a *guide* for self-study, but I don't see why you wouldn't just use something else.

[albin james · Rating 5 out of 5]

To the very end of my dreams