Complex Analysis - Intermediate

Laurent expansions, residues, and the argument principle

University year 2-3

Goals of this section

  • Master the theory and computation of Laurent expansions
  • Understand the classification of singularities (removable, pole, essential)
  • Acquire techniques for computing residues
  • Learn to evaluate complex integrals using the residue theorem
  • Study applications to real integrals (rational, trigonometric, Fourier-type)
  • Understand the argument principle and Rouché's theorem

Prerequisites

  • Introductory material (holomorphic functions, Cauchy's integral theorem and formula)
  • Convergence of series (uniform convergence, absolute convergence)
  • Basics of Taylor expansion

Chapters

Chapter 1: Laurent Expansion

Expanding functions on an annulus; the principal and regular parts of a Laurent series.

Chapter 2: Classification of Singularities

Classifying isolated singularities: removable singularities, poles, essential singularities; Picard's theorem.

Chapter 3: Computing Residues

Definition and computation of residues at simple poles, poles of order n, and essential singularities.

Chapter 4: The Residue Theorem

Proof and applications of the residue theorem; evaluating closed-contour integrals.

Chapter 5: Applications to Real Integrals

Rational, trigonometric, and Fourier-type integrals; Jordan's lemma.

Chapter 6: The Argument Principle

The argument principle, Rouché's theorem, and the open mapping theorem; counting zeros and poles.

Chapter 7: The Riemann Sphere

The extended complex plane, stereographic projection, and holomorphy at infinity.

Chapter 8: Rouché's Theorem

Rouché's theorem and its applications, the open mapping theorem, and Hurwitz's theorem.

Chapter 9: Exercises

Practice problems to check your understanding at the intermediate level.

Casorati-Weierstrass Theorem

The image of a holomorphic function is dense near an essential singularity. Proof by contradiction and comparison with Picard's great theorem.

Harmonic Conjugate

Constructing a conjugate harmonic function via the Cauchy-Riemann equations; uniqueness, orthogonality of level curves, and the complex potential.

Hadamard Three-Circles Theorem

log M(r) is a convex function of log r. The maximum modulus principle and applications to interpolation theory.

Phragmén-Lindelöf Principle

An extension of the maximum modulus principle to unbounded domains. Growth conditions and reduction via auxiliary functions.

Overview

The intermediate level centers on the residue theorem, the most practical and powerful tool in complex analysis. Cauchy's integral theorem from the introductory level stated that "the contour integral of a holomorphic function is zero"; the intermediate level treats the case where the function has points at which it is not holomorphic (singularities).

The Laurent expansion generalizes the Taylor expansion, expanding a function around a singularity into a series containing negative powers. The coefficient of $(z-a)^{-1}$ in this expansion is the residue, which determines the value of a contour integral.

One striking application of the residue theorem is the evaluation of real integrals. By extending an integral over the reals into the complex plane and computing residues, one can evaluate integrals that are difficult by elementary methods. For example:

$$\displaystyle\int_0^\infty \dfrac{1}{1+x^2}\,dx = \dfrac{\pi}{2}, \quad \displaystyle\int_0^\infty \dfrac{\sin x}{x}\,dx = \dfrac{\pi}{2}$$

These formulas can be derived systematically using the residue theorem.

The argument principle is also a powerful tool that counts the zeros and poles of a function by an integral, and through Rouché's theorem it is applied to existence proofs for the zeros of polynomials and analytic functions.

Reading

The Magic of ResiduesReading

An integral that is hopeless on the real line is settled in an instant once you step into the complex plane. A relaxed tale of the mysterious detour where a single number — the residue — holds the value of the integral.

The Mystery of Path IndependenceReading

No matter which path you take, the integral gives the same answer — a casual tale of the "path-independence" miracle born from Cauchy's integral theorem, set against the real-number world and the effect of holes.

The Magic of Angle-Preserving MapsReading

Stretch it or twist it, and only the tiny angles survive intact. A relaxed tale of how holomorphic functions become "angle-preserving maps," and the idea of recasting hard domains into easy shapes, with a nod to the Mercator projection.
  • Cross-Ratio — definition of the cross-ratio, invariance under Möbius transformations, and applications to concyclic / collinear tests, at the intermediate level.
  • Casorati-Weierstrass Theorem — proof and examples of the density of a holomorphic function's range near an essential singularity.
  • Blaschke Product — definition of the Blaschke product, its relation to bounded holomorphic functions on the unit disk, and the convergence (Blaschke) condition.
  • Mittag-Leffler Theorem — given any set of points and principal parts, a meromorphic function having them as the principal parts of its poles can be constructed.
  • Schwarz-Christoffel Mapping — derivation of the formula conformally mapping the upper half-plane or unit disk onto a polygonal domain, parameter determination, and applications.
  • Weierstrass Factorization Theorem — proof that an entire function with a prescribed sequence of zeros can be written as an infinite product of elementary factors.
  • Jensen's Formula — the formula relating the integral of the log of an analytic function on a disk to the number of its zeros, a bridge to Nevanlinna theory.
  • Schwarz Reflection Principle — analytic continuation across the real axis for holomorphic functions that take real values there.
  • Branch Point — definition of branch points of multivalued functions, the choice of branch cuts, and worked examples for the logarithm and square root.

Frequently Asked Questions

What do you learn in intermediate complex analysis?

Laurent expansions, the classification of singularities (removable singularities, poles, essential singularities), computing residues and the residue theorem, applications of the residue theorem to real integrals, the argument principle and Rouché's theorem, and the Riemann sphere are studied systematically.

Why can the residue theorem be used to evaluate real integrals?

When an integral over the real line is extended to an integral along a closed contour in the complex plane, its value is determined solely by the sum of the residues inside the contour (the residue theorem). By choosing a semicircular or rectangular path and eliminating the contribution of the unwanted parts, even integrals that are difficult by elementary means can be evaluated systematically.

How does a Laurent expansion differ from a Taylor expansion?

A Taylor expansion can only be used around a point where the function is holomorphic, whereas a Laurent expansion allows negative-power terms (the principal part) and so can expand a function on an annulus containing a singularity. The coefficient of $1/(z-a)$ in the principal part is the residue.