Learning #33 – Complex Analysis in Physics & Engineering

Complex Numbers & the Argand Plane

A complex number z = x + iy is a point in the Argand plane, with real part Re(z) = x and imaginary part Im(z) = y. The modulus |z| = √(x² + y²) and argument arg(z) = arctan(y/x) give polar form z = r e^iθ, where Euler’s formula

Euler's formula — the most beautiful equation in mathematics e^iθ = cos θ + i sin θ
e^iπ + 1 = 0 (Euler's identity)
Proof: Taylor series of e^iθ = 1 + iθ − θ²/2! − iθ³/3! + … = cos θ + i sin θ

Multiplication in polar form: z₁ z₂ = r₁r₂ e^{i(θ₁+θ₂)} — moduli multiply, arguments add. De Moivre’s theorem: (e^iθ)ⁿ = e^inθ gives sin(nθ) and cos(nθ) in terms of powers of sin/cos. The n-th roots of unity are z_k = e^2πik/n, equally spaced on the unit circle.

Analytic Functions & the Cauchy-Riemann Equations

A function f(z) = u(x,y) + iv(x,y) is complex differentiable at z₀ if the limit (f(z₀+h) − f(z₀))/h exists as h → 0 from any direction in the complex plane. This imposes a strong constraint: the Cauchy-Riemann equations:

Cauchy-Riemann equations ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x

Consequence: both u and v satisfy Laplace's equation
∇²u = 0 and ∇²v = 0
u and v are called harmonic conjugates

A function differentiable throughout a region is called analytic (or holomorphic) there. Analytic functions have remarkable properties:

Differentiable once ⟹ differentiable infinitely often (unlike real functions!).
Representable as a convergent power series (Taylor series) in a disk.
Determined in the whole connected domain by values on any arc (identity principle).
Their real and imaginary parts are harmonic — they model potential fields.

Common Analytic Functions

Polynomials, e^z, sin z, cos z, ln z (with branch cut) are analytic away from their singularities. Key identities:

sin z = (e^iz − e^−iz)/(2i) | cos z = (e^iz + e^−iz)/2
sinh z = (e^z − e^−z)/2 | sin(iz) = i sinh z
ln z = ln|z| + i arg(z) (multi-valued; branch cut at negative real axis)

Contour Integrals & Cauchy’s Theorem

We integrate complex functions along paths (contours) in the complex plane. Cauchy’s theorem: if f(z) is analytic in a simply connected domain, then ∮_C f(z) dz = 0 for any closed contour C in that domain. This is the complex analogue of path-independence of conservative fields.

Cauchy's Integral Formula f(z₀) = (1/2πi) ∮_C f(z)/(z − z₀) dz (C encloses z₀)

f⁽ⁿ⁾(z₀) = (n!/2πi) ∮_C f(z)/(z − z₀)ⁿ⁺¹ dz

Consequence: values of analytic f inside C determined completely by boundary values

Cauchy’s integral formula is non-trivially powerful: knowing f on the boundary of a disk determines it everywhere inside. This is why analytic functions are so rigid compared to smooth real functions. The formula also gives all derivatives for free, proving that analytic ⟹ C∞.

Singularities, Laurent Series & the Residue Theorem

Where analytic functions fail to be analytic, they have singularities. In an annulus r₁ < |z − z₀| < r₂ around a singularity, f expands as a Laurent series:

Laurent series and residue f(z) = ∑_n=−∞^∞ aₙ (z − z₀)ⁿ

Residue: Res(f, z₀) = a₋₁ (coefficient of (z−z₀)⁻¹)

Types of singularity:
• Removable: principal part (n<0 terms) = 0, singularity fillaable
• Pole of order m: principal part finite, aₙ = 0 for n < −m
• Essential: infinite principal part (e.g. e^1/z at 0)

The Residue Theorem is the workhorse of applied complex analysis:

Residue Theorem ∮_C f(z) dz = 2πi ∑ Res(f, zₖ) (sum over poles inside C)

Also: Res(f, z₀) = lim_z→z₀ [(z−z₀) f(z)] for a simple pole
For pole of order m: Res = (1/(m−1)!) lim d^m−1/dz^m−1 [(z−z₀)ᵐ f(z)]

By choosing contours cleverly (usually a large semicircle closed in the upper or lower half-plane), the residue theorem evaluates real integrals that resist elementary methods. Classic examples:

∫₋∞^∞ dx/(1+x²) = π   (one simple pole at i)
∫₀^∞ sin(x)/x dx = π/2   (Jordan's lemma, keyhole contour)
∫₀^2π dθ/(a + b cos θ) = 2π/√(a²−b²)   (unit circle contour)

Conformal Mappings

An analytic function w = f(z) with f′(z) ≠ 0 is a conformal mapping: it preserves angles between curves. This makes conformal maps the key tool for solving 2D boundary-value problems: map a complicated domain to a simple one (disk, half-plane), solve there, map back.

Möbius Transformations

The family w = (az+b)/(cz+d), ad−bc ≠ 0, maps circles and lines to circles and lines. Any Möbius transformation is a composition of translations, rotations, scalings, and inversion w = 1/z. Three points determine a unique Möbius transformation; cross-ratios are preserved.

Joukowski Transform

The Joukowski map w = z + R²/z turns a circle in the z-plane into an aerofoil in the w-plane. Combined with circulation Γ (Kutta condition), it gives the lift-generating flow around a wing:

Joukowski aerofoil + Kutta-Joukowski lift theorem w = z + c²/z (Joukowski map, c = R for symmetric aerofoil)
Complex potential: F(z) = U(z + R²/z) + (iΓ/2π) ln(z/R)
Lift per unit span: L = ρ U Γ (Kutta-Joukowski theorem)
Γ = 4πUR sin α (for thin symmetric aerofoil at angle of attack α)

Schwarz-Christoffel Mapping

The Schwarz-Christoffel formula maps the upper half-plane (or unit disk) to the interior of a polygon with prescribed vertex angles: dw/dz = K ∏ (z − xₖ)^{(αₖ/π − 1)}, where x₁<x₂<…<xₙ are real preimages of the polygon vertices and αₖ are the interior angles. This underpins calculation of electric fields inside waveguides and around sharp conductor edges.

Physical Applications

2D Irrotational Fluid Flow

For incompressible, irrotational 2D flow, the velocity field derives from a velocity potential φ: v = ∇φ with ∇²φ = 0. Since φ satisfies Laplace, it is the real part of an analytic complex potential F(z) = φ(x,y) + iψ(x,y), where ψ is the stream function (∇φ · ∇ψ = 0: streamlines ⊥ equipotentials). The complex velocity is dF/dz = vₓ − ivᵧ. Conformal maps then transform simple flows (uniform, vortex, source) into arbitrary geometries — the basis of classical aerodynamics before CFD.

2D Electrostatics & Method of Images

The electrostatic potential V satisfies ∇²V = −ρ/ε₀. In charge-free regions ∇²V = 0, so V is harmonic. The method of images places fictitious charges outside the domain to satisfy boundary conditions: a charge q near a grounded plane introduces an image −q at the mirror point. Conformal maps extend this to cylindrical, spherical, and more complex conductor geometries.

AC Circuit Impedance

For sinusoidal signals at angular frequency ω, voltages and currents are represented as complex phasors V = V₀ e^iωt. Impedances are complex: Z_R = R, Z_L = iωL, Z_C = 1/(iωC). The complex impedance Z(ω) encodes both magnitude (|Z|) and phase angle (arg Z) simultaneously. Resonance occurs when Im Z = 0; the power absorbed is Re(V I*)/2. This is why complex numbers are taught in every electrical engineering curriculum.

Fourier & Laplace Transforms, Quantum Mechanics

The Fourier transform F(ω) = ∫ f(t) e^−iωt dt is a complex function of frequency. Poles of F(ω) in the complex ω-plane determine the time-domain behaviour: poles in the lower half-plane → exponentially decaying oscillations (stable). The Laplace transform F(s) = ∫₀^∞ f(t) e^−st dt is used for causal signals; its inverse is a Bromwich contour integral closed in the left half-plane. In quantum mechanics, the Feynman path integral and the propagator K(x,t; x₀,t₀) are naturally expressed via contour integrals, and the poles of the S-matrix in the complex momentum plane correspond to bound states and resonances.

Analytic Continuation & the Riemann Zeta Function

If two analytic functions agree on a curve or open set, they agree everywhere on their common domain of analyticity — this is analytic continuation. It allows extending a function beyond its original domain: the Riemann zeta function ζ(s) = ∑ n^−s converges only for Re(s) > 1, but analytic continuation extends it to all ℂ except s = 1. The Riemann hypothesis (all non-trivial zeros lie on Re(s) = 1/2) is equivalent to the sharpest known estimates for the prime-counting function π(x) and remains one of the Millennium Prize Problems.

Riemann zeta function & functional equation ζ(s) = ∑_n=1^∞ n^−s = ∏_{p prime} (1 − p^−s)⁻¹
Functional equation: ζ(s) = 2ˢ πˢ⁻¹ sin(πs/2) Γ(1−s) ζ(1−s)
Trivial zeros: s = −2, −4, −6, …
Non-trivial zeros: s = ½ + iγₙ (Riemann hypothesis: all have Re(s) = ½)

Key Takeaways

Complex differentiability (analyticity) is far more restrictive than real differentiability: analytic ⟹ C∞, power series, global constraints.
Cauchy-Riemann equations connect complex differentiability to harmonic analysis; real and imaginary parts of analytic functions satisfy Laplace's equation.
The residue theorem turns closed contour integrals into sums of residues, enabling systematic evaluation of real improper integrals.
Conformal mappings preserve angles and transform complicated boundaries into simple ones, solving Laplace-equation problems in 2D fluid flow, electrostatics, and heat conduction.
Complex phasors, Fourier/Laplace transforms, and quantum propagators all exploit the algebraic and geometric power of ℂ.
Analytic continuation gives functions like ζ(s) meanng beyond their original domain; the Riemann hypothesis links ζ zeros to the deep structure of prime numbers.