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A Second Course in Complex Analysis

William A. Veech
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is a very careful treatment of several topics in complex analysis. It covers all the details in the proofs and doesn't do any hand-waving over the tricky parts, It's not in a hurry to get anywhere, and takes a number of digressions. This new Dover edition is a slightly-corrected version of the 1967 work published by W. A. Benjamin.

The first half of the book has a strong geometrical flavor, dealing with analytic continuation, the Riemann mapping theorem, the Koebe uniformization theorem, and the Koebe-Faber distortion theorem. It then makes a smooth transition to the world of special functions, with chapters on the elliptic modular function λ(z) (used to prove Picard's two theorems on exceptional values of entire functions and at essential singularities), the Hadamard product theorem (with a detailed study of the Gamma function), and the Riemann zeta function (including Ikehara's proof of the Prime Number Theorem).

Is this the book for you? That depends on whether you like the choice of topics and the careful and geometric treatment. The topics were already classical in 1967 (as Veech notes in the Preface), and you should not shy away from this book just because it is old, especially since it is bargain-priced. But there are numerous other sources for these results:

  • Much of the material is also in Titchmarsh's classic Theory of Functions, although Titchmarsh takes a much less geometric approach. Titchmarsh gives a more direct proof of Picard's theorems that does not depend on the modular function.
  • The Prime Number Theorem is proved in many number theory books, usually with Ikehara's method, although it is rarely treated in complex analysis books. An exception is Bak and Newman's Complex Analysis (Springer, 2nd edition, 1997) which gives the even simpler proof by D. J. Newman that had not been discovered when Veech's book was written.
  • There are several specialized texts on modular functions, for example Apostol's Modular Functions and Dirichlet Series in Number Theory.
  • The material on the mapping theorems and the distortion theorem is the hardest to find. There was a whole industry of this subject (called schlicht functions) in the early 1900s but it has disappeared today. There is a good treatment as a series of problems in Pólya and Szegö's Problems and Theorems in Analysis , Part IV, Chapter 2.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.


Chapter 1: Analytic Continuation

1. The Exponential Function and the Logarithm
2. Continuation Sequences
3. Continuation Along an Arc
4. Germs
5. Existence of Continuations
6. The Winding Number
7. The Argument Principle
8. The Monodromy Theorem
9. Composition of Germs
10. Composition of Continuations
11. Covering Surfaces

Chapter 2 Geometric Considerations

1. Complex Projective Space
2. Linear Transformations
3. Fractional Linear Transformations
4. Properties of Fractional Linear Transformations
5. Symmetry
6. Schwarz's Lemma
7. Non-Euclidean Geometry
8. The Schwarz Reflection Principle

Chapter 3 The Mapping Theorems of Riemann And Koebe

1. Analytic Equivalence
2. Local Uniform Convergence
3. A Theorem of Hurwitz
4. Implications of Pointwise Convergence
5. Implications of Convergence on a Subset
6. Approximately Linear Functions — Another Application of Schwarz's Lemma
7. A Uniformization Theorem
8. A Closer Look At the Covering
9. Boundary Behavior
10. Lindelöf's Lemma
11. Facts From Topology
12. Continuity at the Boundary
13. A Theorem of Fejér

Chapter 4 The Modular Function

1. Exceptional Values
2. The Modular Configuration
3. The Landau Radius
4. Schottky's Theorem
5. Normal Families
6. Montel's Theorem
7. Picard's Second Theorem
8. The Koebe-Faber Distortion Theorem
9. Bloch's Theorem

Chapter 5 The Hadamard Product Theorem

1. Infinite Products
2. Products of Functions
3. The Weierstrass Product Theorem
4. Functions of Finite Order
5. Exponent of Convergence
6. Canonical Products
7. The Borel-Carathéodory Lemma — Another Form of Schwarz's Lemma
8. A Lemma of H. Cartan
9. The Hadamard Product Theorem
10. The Gamma Function
11. Standard Formulas
12. The Integral Representation of Γ(z)


Chapter 6 The Prime Number Theorem

1. Dirichlet Series
2. Number-Theoretic Functions
3. Statement of the Prime Number Theorem
4. The Riemann Zeta Function
5. Analytic Continuation of ζ(s)
6. Riemann's Functional Equation
7. The Zeros Of ζ(s) In The Critical Strip
8. ζ(s) for Re(s) = 1
9. Integral Representation of Dirichlet Series
10. Integral-Theoretic Lemmas
11. Weak Limits
12. A Tauberian Theorem