Week 11 — Complex Analysis: Analytic Functions, Power Series, and Residues

Overview

Complex analysis is, for this course, the mathematics directly under signal processing and control: the region of convergence, poles and zeros, stability, and the Laplace and z-transforms of Weeks 12–13 are all complex-analysis statements. This week develops complex numbers and the complex plane, what it means for a function to be analytic (the Cauchy–Riemann equations), the remarkably rigid consequences of analyticity (Cauchy’s theorem and integral formula), power and Laurent series, and the residue calculus that evaluates integrals and characterizes poles. It sits deliberately right before the transforms.

Readings

Bak & Newman Ch 1 — Complex numbers. The complex plane, modulus and argument, Euler’s formula.
Bak & Newman Ch 2 — Functions of the complex variable \(z\). Complex functions, limits, continuity.
Bak & Newman Ch 3 — Analytic functions. Differentiability, the Cauchy–Riemann equations, harmonic functions.
Bak & Newman Ch 6 — Power series. Radius of convergence, Taylor series of analytic functions.
Bak & Newman Ch 7 — Laurent series, poles, and residues. Singularities, the residue theorem.

Key Concepts

Complex numbers and the plane

A complex number \(z = x + iy = re^{i\theta}\) lives in the plane with modulus \(r=|z|\) and argument \(\theta\). Euler’s formula

\[e^{i\theta} = \cos\theta + i\sin\theta\]

is the identity that makes complex exponentials the natural language of oscillation and rotation — and the reason Fourier analysis (Weeks 12–13) is built on \(e^{j\omega t}\).

Analytic functions and Cauchy–Riemann

\(f\) is analytic (holomorphic) on a region if it is complex-differentiable there. Writing \(f = u + iv\), differentiability forces the Cauchy–Riemann equations:

\[\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.\]

These couple the real and imaginary parts so tightly that analyticity is far stronger than real differentiability: \(u\) and \(v\) are harmonic, and an analytic function is automatically infinitely differentiable.

Cauchy’s theorem and integral formula

For an analytic \(f\) on a simply connected region, the contour integral around any closed loop vanishes:

\[\oint_\gamma f(z)\,dz = 0,\]

and the value inside is determined by the boundary via the Cauchy integral formula \(f(z_0) = \frac{1}{2\pi i}\oint_\gamma \frac{f(z)}{z - z_0}\,dz\). Analytic functions are rigid: their boundary values determine them everywhere.

Power and Laurent series

An analytic function equals its Taylor series within a disk whose radius of convergence reaches the nearest singularity (the complex version of Week 5’s power-series theory). Around an isolated singularity, the Laurent series adds negative powers:

\[f(z) = \sum_{n=-\infty}^{\infty} c_n (z - z_0)^n.\]

The nature of the singularity — removable, pole (finitely many negative terms), or essential — is read off the negative part. Poles and zeros are exactly the objects that describe a system function \(H(z)\) or \(H(s)\).

Residues, poles, and the link to transforms

The residue of \(f\) at a pole \(z_0\) is the coefficient \(c_{-1}\) of its Laurent series, and the residue theorem

\[\oint_\gamma f(z)\,dz = 2\pi i \sum_k \operatorname{Res}(f, z_k)\]

evaluates contour integrals by summing residues of enclosed poles — the standard tool for inverse Laplace/z-transforms and for evaluating real integrals. This is the precise foundation for Week 13: a transfer function’s poles govern stability (left half-plane for Laplace, inside the unit circle for the z-transform), its zeros shape the response, and the region of convergence is the annulus where the Laurent/transform series converges.

Connections

Backward: complex power series extend the uniform-convergence and radius-of-convergence theory of Week 5; the plane geometry uses the inner-product/rotation ideas of Weeks 1–2.
Forward: Weeks 12–13 apply poles, zeros, ROC, and residues directly to the Laplace and z-transforms, stability, and inverse transforms; the same analytic machinery reappears in Fourier optics (Week 18).
Across courses: filter and control-loop stability, transform-domain reasoning for DSP and sensor front ends (Courses 1, 4).