Complex Analysis

Citation

Stewart, I., & Tall, D. (2018). Complex Analysis (2nd ed.). Cambridge University Press.

Chapter Summary

The book has eighteen numbered chapters, starting at Chapter 0, in roughly 380 pages. Cambridge University Press describes it as developing complex analysis from the established theory of real analysis, emphasising the differences that come from the richer geometry of the complex plane, and using simple topological ideas to turn visual intuition into rigorous proof. The second edition keeps the corrected original text and adds three chapters: 15 (infinitesimals), 16 (the homology version of Cauchy's theorem) and 17 (further directions).

Chapter 0: The Origins of Complex Analysis, and Its Challenge to Intuition

A historical opening of twelve pages: the origins of complex numbers, controversies about their meaning, their representation in the plane, Cauchy's use of infinitesimals in the nineteenth century, and Weierstrass's arithmetization of analysis, which underlies the formal approach of the first edition. The second edition adds a new formal approach based on an extension field containing infinitesimals, and the chapter raises the relationship between visual intuition and rigour.

Chapters 1–2: Algebra of the Complex Plane; Topology of the Complex Plane

Chapter 1 defines complex numbers as ordered pairs of reals with addition and multiplication, and covers real and imaginary parts, modulus, conjugates and polar coordinates, including a proof that the complex numbers cannot be ordered. Chapter 2, the longest of the early chapters at 35 pages, covers open and closed sets, limits and continuity, paths, connectedness and domains, and proves the Paving Lemma: a path in a domain can be covered by finitely many discs, so any two points of the domain can be joined by a path of horizontal and vertical steps.

Chapters 3–5: Power Series; Differentiation; The Exponential Function

Chapter 3 extends convergence from real to complex sequences and series: Cauchy's general principle of convergence, absolute convergence, the comparison and ratio tests, and power series in a disc. Chapter 4 defines the derivative by analogy with the real case and shows that power series are infinitely differentiable inside their circle of convergence, contrasting this with the "bad behaviour" of real functions. Chapter 5 defines exp, sin and cos as power series and proves Euler's formula, periodicity and the related trigonometric and hyperbolic functions.

Chapters 6–7: Integration; Angles, Logarithms, and the Winding Number

Chapter 6 treats complex integration in detail: contours, the integral along a path, the fundamental theorem of contour integration and the Estimation Lemma, which is used in later proofs. Chapter 7 deduces the argument from real power series, defines the winding number of a path round a point through a continuous choice of argument and as an integral, and discusses the components of the complement of a closed path.

Chapters 8–9: Cauchy's Theorem; Homotopy Versions of Cauchy's Theorem

Chapter 8 proves Cauchy's theorem for a triangle, for a star domain, and for any path that does not wind round a point outside the domain, using local antiderivatives and integration around nearby step paths; it also discusses cuts, Jordan contours and simply connected domains. Chapter 9 gives homotopy versions for boundaries and for homotopic paths with fixed endpoints or closed paths, all resting on the local existence of an antiderivative and the fundamental theorem of contour integration.

Chapters 10–12: Taylor Series; Laurent Series; Residues

Chapter 10 proves Taylor series from Cauchy's integral formula and gets Morera's theorem, Cauchy's estimate, Liouville's theorem and a proof of the fundamental theorem of algebra, together with zeros, extension of functions and local extrema of the modulus. Chapter 11 introduces Laurent series, isolated and essential singularities, the Riemann sphere and behaviour at infinity. Chapter 12 covers the residue theorem and its use for infinite integrals, infinite series and other exercises.

Chapters 13–14: Conformal Transformations; Analytic Continuation

Chapter 13 treats conformal properties of functions where the derivative is non-zero, shows that Möbius functions preserve circles, and applies conformal maps to two-dimensional potential theory, Laplace's equation (through the Cauchy–Riemann equations) and the design of aerofoils. Chapter 14 explains how a power series limited to a disc can be extended through overlapping domains, which leads to multiform functions and Riemann surfaces, with the logarithm, the square root and complex powers as examples.

Chapters 15–17: Infinitesimals in Real and Complex Analysis; Homology Version of Cauchy's Theorem; The Road Goes Ever On …

The three chapters new to the second edition. Chapter 15 (35 pages) gives a formal foundation for an extension of the real and complex numbers containing infinitesimals, visualised by magnification, and relates it to classical, standard and non-standard analysis. Chapter 16 extends the homotopy versions of Cauchy's theorem by asking how holes caused by singularities affect integrals; using step paths from the Paving Lemma it defines cycles and boundaries from sums of rectangles and shows that the integral round a boundary is zero. Chapter 17 is an eight-page outline of where the subject has gone and is going: the Riemann Hypothesis, modular functions, several complex variables, complex manifolds, complex dynamics and fractal geometry.

Key Concepts

Chapters 0–5: Foundations

• Complex numbers as ordered pairs: the formal definition of Chapter 1, in which multiplication corresponds to defining i squared as minus one.
• Paving Lemma: Chapter 2's tool for turning pictures of paths into step paths, used again in Chapters 7 and 16.
• Infinite differentiability of complex functions: Chapter 4's contrast with real functions.
• Euler's formula: exp(z) = cos(z) + i sin(z), proved in Chapter 5 from the power-series definitions.

Chapters 6–14: Integration and the main theorems

• Estimation Lemma: bounds a contour integral by contour length and the maximum of the integrand, and is used in later proofs.
• Winding number: Chapter 7's measure of how often a path goes round a point.
• Cauchy's theorem and its versions: for a triangle, a star domain, homotopic paths (Chapters 8–9) and boundaries (Chapter 16).
• Residues, conformal maps and Riemann surfaces: Chapters 12–14.

Chapters 15–17: Second-edition additions

• Infinitesimals: the extension field of Chapter 15.
• Cycles and boundaries: the homology language of Chapter 16.

Critical Analysis

Strengths

• A consistent method. Topology is developed first (Chapter 2) and its Paving Lemma reappears in the winding number (Chapter 7) and the homology theorem (Chapter 16), so intuition about paths becomes a proof technique rather than a hand-wave. The publisher highlights exactly this strategy.
• Cauchy's theorem treated in depth. The theorem appears in three guises: for triangles and star domains with cuts and Jordan contours (Chapter 8), via homotopy (Chapter 9), and via homology (Chapter 16), each building on local antiderivatives. Students can see how the hypotheses can be weakened step by step.
• History and motivation. Chapter 0 gives the historical background and the intuition-versus-rigour theme before any technical work.
• Applications in the book. Chapter 12 uses residues to evaluate infinite integrals and series, and Chapter 13 applies conformal maps to potential theory and aerofoil design.

Limitations

• A long wait for complex differentiation. Differentiation begins on page 75, after a 35-page topology chapter, so students who want to see analytic functions quickly will find the opening slow; the topology is the price of the rigorous route.
• Real analysis is assumed. The book builds from real analysis, so it suits students who have already met limits, sequences and series rigorously.
• Non-standard material. Chapter 15 gives a formal theory of infinitesimals, a perspective that departs from most undergraduate courses; readers following a standard syllabus can skip it, as they can the short Chapter 17.
• Limited applications. The applications are chiefly residue calculations and conformal mapping for potential theory; there is no chapter on, for example, Fourier or Laplace transforms.

Where It Fits in the Course

Stewart and Tall sit between the gentler introductions and the graduate classics in this course list. Beck, Marchesi, Pixton and Sabalka's A First Course in Complex Analysis is a free one-semester text that relies on as little real analysis as possible and works towards the residue theorem; Brown and Churchill's Complex Variables and Applications is an introduction that keeps applications in view. Either is a better first meeting with the subject for students without a real analysis course. Needham's Visual Complex Analysis is the natural companion for geometric intuition, covering the same ideas through pictures and physical intuition rather than through formal proof alone. Ahlfors's Complex Analysis and Stein and Shakarchi's Complex Analysis are concise graduate texts to move to afterwards.

How to Use This Book

The book is meant for undergraduates who have completed a first course in real analysis and multivariable calculus and who want a proof-based but geometrically motivated treatment of complex analysis.

A practical plan:

• Core (Chapters 1–12): read in order. Chapter 2 is the most unfamiliar for a first-time reader, but its Paving Lemma pays off in Chapters 7 and 8. Chapters 6–12 carry the central theorems: integration, Cauchy's theorem and its homotopy version, Taylor and Laurent series, and residues.
• Applications and continuation (Chapters 13–14): conformal maps suit anyone interested in physics or engineering, and analytic continuation leads to Riemann surfaces.
• Optional (Chapters 0, 15–17): Chapter 0 is a good read for context at any point; Chapter 16 offers a deeper, homology version of Cauchy's theorem for a second pass; Chapter 15 suits readers interested in infinitesimals; Chapter 17 is a short guide to further reading.

Work the exercises alongside the text, especially the residue exercises of Chapter 12, which put the central theorems to use.