A First Course in Complex Analysis

Citation

Beck, M., Marchesi, G., Pixton, D., & Sabalka, L. (2025). A First Course in Complex Analysis (Edition 1.6). Orthogonal Publishing L3C. Free official edition: https://matthbeck.github.io/complex.html

Chapter Summary

Chapter 1: Complex Numbers

After a brief historical note on how complex numbers arose from Cardano's formula for cubics, the chapter defines ℂ formally as pairs of real numbers with a particular multiplication and proves that it is a field. It then moves between algebraic and geometric viewpoints—polar form, the exponential notation e^{iφ}, roots of unity, conjugates, absolute value and the triangle inequality—and closes with the elementary topology of the plane (open and closed sets, boundaries, regions, paths and connectedness) that later chapters need. An optional lab asks students to compute with complex numbers by hand and check the results in SageMath.

Chapter 2: Differentiation

Limits and continuity are defined for complex functions, and complex differentiability is introduced; a function differentiable on an open set is called holomorphic. The Cauchy–Riemann equations are derived as the condition linking complex differentiability to the partial derivatives of the real and imaginary parts, harmonic functions make a first appearance, and a short section proves that a function with zero derivative on a region is constant.

Chapter 3: Examples of Functions

This chapter builds a stock of examples. Möbius transformations are shown to map circles and lines to circles and lines; adding a point at infinity leads to the cross ratio and to stereographic projection onto the sphere. The exponential and trigonometric functions are extended to complex arguments, where they acquire new features such as periodicity of the exponential, and the logarithm is treated through branches, along with complex powers.

Chapter 4: Integration

Complex integrals are defined along piecewise smooth paths, with basic properties and the standard estimate in terms of path length. The chapter studies antiderivatives and when integrals are path-independent, introduces homotopy of closed paths, and proves Cauchy's Theorem in a homotopy form. It ends with Cauchy's Integral Formula, first for disks (including a mean-value version) and then for more general simple closed paths.

Chapter 5: Consequences of Cauchy's Theorem

Building on Chapter 4, this chapter derives integral formulas for derivatives and concludes that a holomorphic function is infinitely differentiable. Morera's Theorem and the existence of antiderivatives on simply connected regions follow, and the final section "takes Cauchy's formulas to the limit" to prove the Fundamental Theorem of Algebra and Liouville's Theorem on bounded entire functions.

Chapter 6: Harmonic Functions

Returning to real-valued functions, the chapter shows that real parts of holomorphic functions are harmonic and that, on simply connected regions, every harmonic function is such a real part. It proves the mean-value property and the maximum–minimum principle for harmonic functions, and discusses the Dirichlet problem and the Poisson kernel through the exercises.

Chapter 7: Power Series

Motivated by an integral that the earlier tools cannot evaluate, this chapter develops sequences, completeness, series and convergence tests from scratch, then sequences and series of functions with uniform convergence. It proves that a power series has a radius of convergence and describes where such series converge.

Chapter 8: Taylor and Laurent Series

The chapter connects power series to holomorphic functions in both directions: power series are holomorphic inside their disk of convergence, and every holomorphic function has a local power series expansion. It classifies zeros, proves the identity principle and the maximum-modulus theorem, and introduces Laurent series on annuli, finally computing the motivating integral from Chapter 7 by expanding the integrand around its singularities.

Chapter 9: Isolated Singularities and the Residue Theorem

Isolated singularities are classified as removable, poles or essential, with the Casorati–Weierstrass theorem describing behaviour near an essential singularity. Residues are defined and computed by several rules, leading to the Residue Theorem, which turns integrals around closed paths into sums of residues. The chapter ends with the Argument Principle and Rouché's Theorem for counting zeros and poles, and its exercises apply residues to real improper integrals.

Chapter 10: Discrete Applications of the Residue Theorem

The last chapter is a set of guided, multi-step projects that use complex integration to answer discrete questions: evaluating infinite sums such as the sum of 1/k², identities for binomial coefficients, a formula for Fibonacci numbers, and reciprocity for Dedekind sums. The note to instructors suggests these as longer homework problems or group projects.

Appendix A: Theorems from Calculus

A short appendix collects the real-calculus results used in the text, such as the extreme-value theorem, the mean-value theorem and the fundamental theorem of calculus.

Key Concepts

Chapter 1: Complex Numbers

• Complex numbers as a field: pairs of reals with a defined addition and multiplication satisfying the usual algebraic laws.
• Polar form: writing z = r e^{iφ}, which turns multiplication into multiplying lengths and adding angles.
• Region: an open, connected subset of the plane, the standard domain for complex functions in the book.

Chapter 2: Differentiation

• Holomorphic function: a function that is complex differentiable at every point of an open set.
• Cauchy–Riemann equations: the partial differential equations u_x = v_y and u_y = −v_x satisfied by the real and imaginary parts of a differentiable function.

Chapter 3: Examples of Functions

• Möbius transformation: a map (az + b)/(cz + d) with ad − bc ≠ 0, which sends circles and lines to circles and lines.
• Stereographic projection: a correspondence between the extended plane and the sphere that carries circles to circles.
• Branch of the logarithm: a continuous choice of logarithm on a region, needed because the complex exponential is periodic.

Chapter 4: Integration

• Path integral: integration of a complex function along a parametrised path.
• Cauchy's Theorem: the integral of a holomorphic function over a closed path is zero when the path can be deformed to a point within the region (homotopy version).
• Cauchy's Integral Formula: values of a holomorphic function inside a closed path are determined by an integral over the path.

Chapter 5: Consequences of Cauchy's Theorem

• Infinite differentiability: a holomorphic function has derivatives of all orders, given by integral formulas.
• Liouville's Theorem: every bounded entire function is constant.
• Fundamental Theorem of Algebra: every nonconstant polynomial has a complex root.

Chapter 6: Harmonic Functions

• Harmonic function: a real function with continuous second partial derivatives satisfying Laplace's equation.
• Maximum principle: a nonconstant harmonic function on a region has no interior maximum or minimum.

Chapter 7: Power Series

• Uniform convergence: convergence of a sequence of functions at a rate independent of the point, which preserves continuity and allows integration term by term.
• Radius of convergence: the radius of the disk on which a power series converges.

Chapter 8: Taylor and Laurent Series

• Taylor expansion: every holomorphic function is locally the sum of a power series.
• Identity principle: two holomorphic functions on a region that agree on a set with an accumulation point agree everywhere.
• Laurent series: a two-sided series representing a function holomorphic on an annulus.

Chapter 9: Isolated Singularities and the Residue Theorem

• Residue: the coefficient of (z − z₀)⁻¹ in the Laurent expansion at an isolated singularity.
• Residue Theorem: an integral over a closed path equals 2πi times the sum of the residues enclosed.
• Rouché's Theorem: a comparison criterion showing that two functions have the same number of zeros inside a path.

Chapter 10: Discrete Applications of the Residue Theorem

• Summation by residues: evaluating series such as Σ 1/k² by integrating a function built from π cot(πz) over expanding paths.
• Dedekind sums: number-theoretic sums whose reciprocity law can be proved with the Residue Theorem.

Critical Analysis

Strengths

• Written for a first rigorous course. The authors say that for many of their students complex analysis is the first rigorous analysis course, and the book reflects this: it relies on as little real analysis as possible, develops sequences and series from scratch, and collects the needed calculus theorems in an appendix.
• A clear goal that shapes the structure. The book works toward the Residue Theorem. An integral that the tools of Chapters 4–5 cannot handle is introduced at the start of Chapter 7, computed laboriously with Laurent series in Chapter 8, and dispatched by residues in Chapter 9, so readers see why the series machinery is needed.
• Discrete applications. Chapter 10's guided projects on infinite sums, binomial coefficients, Fibonacci numbers and Dedekind sums are unusual in an introductory text and show the Residue Theorem linking continuous and discrete mathematics.
• Exercises and software. More than 250 exercises, many with hints, and numerous SageMath prompts are spread through the text; the open licence (CC BY 4.0) lets instructors adapt it.

Limitations

• Power series arrive late. Because series are built from scratch, Taylor series appear only in Chapter 8. Students who already know real analysis may find Chapter 7 slow, and instructors who prefer a power-series-first approach will need to reorder material.
• Limited coverage beyond residues. Conformal mapping is limited to Möbius transformations and the elementary functions; the Riemann mapping theorem, analytic continuation, infinite products and the Gamma and zeta functions are not covered. Further study requires a more advanced text.
• Applications are mostly mathematical. Apart from the motivation for harmonic functions and the Dirichlet problem, there is little on physical or engineering applications such as fluid flow or transforms.
• Real integrals are left to exercises. The standard evaluation of real improper integrals by residues appears only in exercises rather than worked sections, so students who need this technique for applied courses need extra examples.

Applications in the Book

Applications are concentrated in Chapter 10 and in the exercises. Chapter 10 leads readers through computing ζ(2) and an alternating analogue via residues of π cot(πz)/z², identities for binomial coefficients, a closed formula for the Fibonacci numbers, and the reciprocity law for Dedekind sums (an approach the authors credit to Rademacher). Exercises in Chapters 5 and 9 evaluate improper real integrals of rational functions, and Chapter 6 connects harmonic functions to the Dirichlet problem and the Poisson kernel.

Where It Fits in the Course

In this Complex Analysis list, Beck and colleagues' book sits alongside Brown and Churchill's Complex Variables and Applications, which is generally described as an applications-oriented introduction; Beck et al. is the more proof-centred of the two, while Brown and Churchill is commonly used where engineering applications matter. Tristan Needham's Visual Complex Analysis is generally described as a geometric, picture-driven treatment and works well as a companion for intuition, especially for Chapter 3's Möbius transformations and the geometry of complex multiplication. Stewart and Tall's Complex Analysis is another undergraduate-level option.

Ahlfors's Complex Analysis and Stein and Shakarchi's Complex Analysis are generally used for graduate or honours courses. Beck et al. is good preparation for either: its homotopy form of Cauchy's Theorem, residue calculus and Rouché's Theorem form the base on which those texts build conformal mapping, analytic continuation and special functions.

How to Use This Book

The book is aimed at undergraduates taking complex analysis as perhaps their first rigorous analysis course. Readers need multivariable calculus (partial derivatives, line-integral ideas, polar coordinates) and some comfort reading and writing proofs; real analysis is not assumed.

The authors note that there is more material than fits in a semester. A core path is Chapters 1–5, then Chapters 7–9, culminating in the Residue Theorem; Chapter 6 on harmonic functions and parts of Chapter 3 (stereographic projection, cross ratio) can be shortened, and students who know real analysis can move quickly through Sections 7.1–7.2. Chapter 10 works best as end-of-course projects. Work the exercises at the end of each chapter—hints are given for many—and use the SageMath prompts and the optional lab in Chapter 1 to check computations and visualise functions.

Edition and Sources

This guide is based on Edition 1.6 (August 2025) of A First Course in Complex Analysis (© 2002–2025 Matthias Beck, Gerald Marchesi, Dennis Pixton and Lucas Sabalka, CC BY 4.0), read in the free PDF from the official book page, https://matthbeck.github.io/complex.html; a paperback is published by Orthogonal Publishing. The guide draws on "About This Book", the note to instructors, the full table of contents, the introductions of all ten chapters and the appendix, every section heading, the statements of the main numbered theorems and propositions, selected worked examples (Möbius transformations, homotopy, residue computations) and the exercise sets of Chapters 1, 5, 6 and 9.