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Theory of Functions of a Complex Variable, Volume 1

C. Caratheodory
American Mathematical Society/Chelsea
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This is an idiosyncratic introduction to complex variables, and represents the author’s take on the subject after a lifetime working in the field. The book was originally published in German as Funktionentheorie in 1950, and almost immediately translated into English and published by Chelsea in 1954. The book was written as a text (although without exercises), but today it works better as a monograph, where you can pick and choose the interesting topics.

The first volume works along a leisurely path from the very simple and general to the very specific. It starts with properties of the complex numbers, then of the complex plane and the Riemann sphere. Then it starts working on some simple transformations of the sphere, specifically inversions and Möbius transforms, which are covered in detail, then there’s a little bit of topology and contour integration, and we finally start looking at analytic functions. Then there’s a good bit about Carathéodory’s theory of continuous convergence (an alternative approach to uniform convergence). Finally, about 3/4 of the way through the first volume, we get to power series and detailed studies of several special functions.

The second volume is more specialized and was likely intended to be a second course after the basics were covered. The first half of the book covers a number of ideas in geometric function theory. The second half covers some triumphs of circa-1900 function theory, such as Picard’s theorem on exceptional values of entire functions (a non-constant entire function omits taking on at most one finite value).

The book has a strong geometric slant, and the topics (although not the treatment) overlap a lot of Ahlfors’s Complex Analysis. It is not very similar in topics or treatment to Needham’s very modern geometric book Visual Complex Analysis, which really is an introductory textbook.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

Part One: Complex Numbers

  • The complex numbers from the algebraic point of view
  • The geometry of the complex numbers
  • Euclidean, spherical, and non-Euclidean geometry

Part Two: Some Results From Point Set Theory and From Topology

  • Convergent sequences of numbers and continuous complex functions
  • Curves and regions
  • Contour integration

Part Three: Analytic Functions

  • Foundations of the theory
  • The maximum-modulus principle
  • The Poisson integral and harmonic functions
  • Meromorphic functions

Part Four: Analytic Functions Defined By Limiting Processes

  • Continuous convergence
  • Normal families of meromorphic functions
  • Power series
  • Partial-fraction decomposition and the calculus of residues

Part Five: Special Functions

  • The exponential and trigonometric functions
  • The logarithmic function and the general power function
  • The Bernoulli numbers and the gamma function
  • Index