Complex Analysis

Citation

Stein, E. M., & Shakarchi, R. (2003). Complex Analysis. Princeton University Press. Princeton Lectures in Analysis, II.

Chapter Summary

The book has ten chapters and two appendices. It is the second of four volumes in the Princeton Lectures in Analysis, and the introduction refers back to the first volume on Fourier analysis. Each chapter is followed by exercises and, in all chapters after the first, by harder problems.

Chapter 1: Preliminaries to Complex Analysis

The complex plane, convergence and sets in the plane, continuous and holomorphic functions, power series, and integration along curves. The chapter prepares the ground for Cauchy's theorem and ends with exercises only.

Chapter 2: Cauchy's Theorem and Its Applications

Goursat's theorem leads to the local existence of primitives and Cauchy's theorem in a disc, first for simple "toy contours". The chapter then evaluates some integrals, proves the Cauchy integral formulas, and gives further applications: Morera's theorem, sequences of holomorphic functions, functions defined by integrals, the Schwarz reflection principle and Runge's approximation theorem.

Chapter 3: Meromorphic Functions and the Logarithm

Zeros and poles, the residue formula with examples, singularities and meromorphic functions, the argument principle and its applications, homotopies and simply connected domains, the complex logarithm, and a closing section relating Fourier series and harmonic functions.

Chapter 4: The Fourier Transform

The action of the Fourier transform on a particular class of functions (denoted ℑ in the contents), together with the Paley–Wiener theorem. The introduction presents this as the elegant link between the Fourier transform and complex analysis.

Chapter 5: Entire Functions

Jensen's formula, functions of finite order, infinite products (including the product formula for the sine), Weierstrass infinite products and Hadamard's factorization theorem.

Chapter 6: The Gamma and Zeta Functions

The gamma function with its analytic continuation and further properties, then the zeta function with its functional equation and analytic continuation.

Chapter 7: The Zeta Function and Prime Number Theorem

Zeros of the zeta function and estimates for 1/ζ(s), then the reduction of the prime number theorem to asymptotics for the functions ψ and ψ₁, with a proof of those asymptotics.

Chapter 8: Conformal Mappings

Conformal equivalence, the disc and upper half-plane, the Dirichlet problem in a strip, the Schwarz lemma and automorphisms, the Riemann mapping theorem (with Montel's theorem), and conformal maps onto polygons through the Schwarz–Christoffel integral, ending with a return to elliptic integrals.

Chapter 9: An Introduction to Elliptic Functions

Elliptic functions, Liouville's theorems and the Weierstrass ℘ function, then the modular character of elliptic functions and Eisenstein series, including their connection with divisor functions.

Chapter 10: Applications of Theta Functions

The product formula for the Jacobi theta function and its transformation laws, generating functions and a short introduction to partitions, and proofs of the two-squares and four-squares theorems. The chapter opening stresses the theta function's dual nature: elliptic in one variable, modular in the other.

Appendices A and B: Asymptotics; Simple Connectivity and Jordan Curve Theorem

Appendix A covers Bessel functions, Laplace's method and Stirling's formula, the Airy function and the partition function. Appendix B gives equivalent formulations of simple connectivity, the Jordan curve theorem, and a proof of a general form of Cauchy's theorem.

Key Concepts

Chapters 1–3: Holomorphic Functions, Cauchy and Residues

• Holomorphic function: a function differentiable in the complex sense, whose condition on the limit of the difference quotient drives the whole theory.
• Cauchy's theorem and integral formulas: the integral of a holomorphic function over a suitable closed curve vanishes, and values inside a circle are given by an integral over it.
• Residue formula and argument principle: tools that turn integrals and counting of zeros and poles into local computations.

Chapters 4–7: Fourier, Entire Functions, Zeta

• Paley–Wiener theorem: the chapter's closing result, relating properties of a function to those of its Fourier transform.
• Hadamard factorization: representing an entire function of finite order through its zeros by an infinite product.
• Functional equation of ζ: the relation that gives analytic continuation and underlies the prime number theorem proof.

Chapters 8–10: Conformal Maps, Elliptic and Theta Functions

• Riemann mapping theorem: every simply connected proper open subset of the plane is conformally equivalent to the disc.
• Weierstrass ℘ function: a basic doubly periodic function from which elliptic functions are built.
• Theta function: a series with elliptic and modular behaviour, used to prove sum-of-squares theorems.

Critical Analysis

Strengths

• A clear opening statement of method. The introduction names three phenomena (contour integration, regularity, analytic continuation) and says the proofs of the main properties are short and illuminating, which gives a reader an overview before the technical chapters.
• Reach beyond a standard syllabus. The publisher's description stresses connections to Fourier analysis, number theory and combinatorics. The table of contents backs this with chapters on the Fourier transform, the prime number theorem, elliptic functions and the sums of squares.
• Practice built into every chapter. After Chapter 1, every chapter ends with exercises and a separate set of problems, and the publisher has posted an errata list for several exercises.
• Rigor with motivation. The publisher describes a careful balance between conceptual insight and the technical underpinnings of rigorous analysis, and chapters open with an epigraph and a statement of what the chapter will do (as in Chapters 2 and 10).

Limitations

• Written as part of a series. The introduction cites results from the first volume, so a reader without Fourier series and some real analysis will meet gaps.
• Little help for first encounters. Chapter 1 is a preliminary chapter ending in exercises, so students new to complex numbers will want a gentler text beside it.
• Specialized later chapters. Chapters 9 and 10 on elliptic and theta functions, and Appendix A on asymptotics, serve readers heading toward number theory and are not needed for a general course.

Where It Fits in the Course

Stein and Shakarchi is a rigorous second-level treatment that connects complex analysis to the rest of analysis. Earlier or gentler options in the list include Beck, Marchesi, Pixton and Sabalka's free one-semester text, Brown and Churchill's introduction, and Stewart and Tall's British undergraduate text. Needham's Visual Complex Analysis explains the subject geometrically and is a good companion for intuition.

Ahlfors's classic graduate text covers the same core at graduate level and is a natural comparison for readers who want a second source. Stein and Shakarchi stands out by integrating Fourier analysis, the zeta function and theta functions into the course, so it is the one to choose for a student who has taken real analysis and wants those applications.

How to Use This Book

The book suits students with a solid real analysis course who have seen, or are ready to learn, Fourier series. It can be used as the main text for a one-semester course or for self-study.

• Core: Chapters 1 to 3 give the central theory (Cauchy, residues, the argument principle, the logarithm). Chapter 8 on conformal mappings is the natural next topic for a general course.
• Next: Chapters 5 to 7 for entire functions and the zeta function lead to the prime number theorem;
• Optional: Chapters 9 and 10 and the appendices for readers interested in elliptic functions, number theory or asymptotics.

Work the exercises as you go, and attempt the problems after each chapter, which are the more demanding items. Consult the publisher's errata for the listed exercises.