Mathematics · Upper Division

Complex Analysis

Complex analysis is the most powerful tool in a physicist's mathematical arsenal. Contour integration evaluates impossible-looking real integrals; conformal maps solve 2D electrostatics; analytic continuation ties together special functions.

Prerequisites

Calculus (Ch. 22) \cdot Differential equations (Ch. 23) \cdot Basic complex numbers

Learning Goals

Verify analyticity using the Cauchy-Riemann equations and identify poles, branch cuts, and essential singularities.
Apply the residue theorem to evaluate real integrals by closing a contour in the complex plane.
Compute residues at simple and higher-order poles using the limit formula.
Derive the Kramers-Kronig relations from causality and explain why they connect absorption to dispersion.
Use La $urent series to classify singularities and extract the residue from the coefficient of (z-$

CA.1 Analytic Functions and the Cauchy-Riemann Equations

A function f(z) = u(x,y) + iv(x,y) is analytic (holomorphic) at a point if it is complex-differentiable in a neighborhood. The necessary and sufficient condition is the Cauchy-Riemann equations:

\partialu/\partialx = \partialv/\partialy \qquad \partialu/\partialy = -\partialv/\partialx \qquad (Cauchy-Riemann)

(CA.1)

Analytic functions are remarkable: they are infinitely differentiable, their real and imaginary parts are both harmonic (∇²u = ∇²v = 0), and they define conformal maps— angle-preserving transformations. Conformal maps reduce Laplace's equation in complicated domains to simple ones.

Key analytic functions and their singularities: e^z (entire), sin z (entire), 1/z (simple pole at z=0), ln z (branch cut), z^(1/2) (branch point), 1/(z²+1) (poles at ±i).

CA.2 The Residue Theorem

Theorem CA.1 — Cauchy's Residue Theorem

For a f

unction f(z) analytic inside and on a closed contour C except at isolated singularities z_{1}

..,

z_{n}

\oint_C f(z) dz = 2\pii \sum_{k} Res(f, z_{k}

where the residue at a simple pole z_{0} is Res(f, z_{0}) = lim(z\toz_{0}) (z-z_{0})f(z). For a pole of

order n: Res(f, z_{0}) = (1/(n-1)!) lim(z\toz_{0}) d^{n-1}/dz^{n-1} [(z-z_{0})^{n}f(z

The residue theorem converts contour integrals (around closed paths in ℂ) into a sum of local quantities (residues) — a spectacular global-from-local result.

Example CA.1 — Evaluating a Real Integral by Contour Integration

$Evaluate I = \int(-\infty to \infty) dx/(1+x^{4}$ .

Poles of 1/(1+z⁴):

z^{4} = -1 = e^{i\pi} \to z = e^{i\pi/4}, e^{i3\pi/4}, e^{i5\pi/4}, e^{i7\pi/4}. Upper half-plane poles

z_{1} = e^{i\pi/4} = (1+i)/\sqrt2, z_{2} = e^{i3\pi/4} = (-1+i)/\sqrt2

Close contour:

Take semicircle in upper half-plane. As R\to\infty, the arc contribution \to 0 (Jordan's lemma: |z^{-}

^{4}| \to 0 fast enough

Residue at z₁:

Res(z_{1}) = 1/(4z_{1}^{3}) = z_{1}/(4z_{1}^{4}) = z_{1}/(4\times(-1)) = -z_{1}/4 = -(1+i)/(4\sqrt2

Residue at z₂:

Res(z_{2}) = -z_{2}/4 = (1-i)/(4\sqrt2

Sum of residues:

\sum Res = [-(1+i) + (1-i)]/(4\sqrt2) = -2i/(4\sqrt2) = -i/(2\sqrt2

Result:

I = 2\pii \times (-i/(2\sqrt2)) = \pi/\sqrt2. ✓ (Verify: \int dx/(1+x^{4}) = \pi/(2sin(\pi/4)) \times 1/2 = \pi/\sqrt2

CA.3 Dispersion Relations and Kramers-Kronig

In physics, causality forces the real and imaginary parts of a response function χ(ω) (susceptibility, refractive index, dielectric function) to be related by the Kramers-Kronig relations:

\chi'(\omega) = (1/\pi) P \int(-\infty to \infty) \chi''(\omega')/(\omega'-\omega) d\omega' \qquad \chi''(\omega) = -(1/\pi) P \int \chi'(\omega')/(\omega'-\omega) d\omega'

(CA.2)

Here P denotes the Cauchy principal value, and χ = χ' + iχ''. These follow from analyticity of χ(ω) in the upper half-plane (causality) and Jordan's lemma via the residue theorem. The relations connect absorption (Im part, χ'') to dispersion (Re part, χ') — you can measure one and compute the other.

CA.4 Laurent Series and Asymptotic Expansions

In an annular region around an isolated singularity z₀, any analytic function has aLaurent series:

f(z) = \sum(n=-\infty to \infty) a_{n} (z-z_{0})^{n} \qquad a_{-1} = Res(f, z_{0})

(CA.3)

The residue is the coefficient of (z−z₀)^(−1). The principal part(negative powers) classifies singularities: finite number of negative powers → pole; infinitely many → essential singularity (e.g., e^(1/z) at z=0).

For large |z|, functions often have useful asymptotic expansions (not necessarily convergent, but useful term-by-term). Example: the Gamma function satisfiesStirling's approximation:

ln Γ(z) \approx (z - \frac{1}{2}) ln z - z + \frac{1}{2} ln(2\pi) + 1/(12z) - \cdots \qquad (Stirling)

(CA.4)

Example CA.2 — Green's Function via Contour Integration

$Find the retarded Green's function for the harmonic oscillator: (d^{2}/dt^{2} + \omega_{0}^{2})G(t) = \delta(t$ .

Fourier transform:

G̃(\omega) = -1/(\omega^{2} - \omega_{0}^{2}). Poles at \omega = \pm\omega_{0}

Retarded condition:

Shift poles below real axis: \omega \to \omega \pm i\varepsilon. G_{ret}(\omega) = -1/((\omega+i\varepsilon)^{2}-\omega_{0}^{2}

Inverse transform:

G(t) = (1/2\pi) \int G̃(\omega) e^{-i\omegat} d\omega

For t > 0:

Close in lower half-plane. Both poles now inside. Sum residues: G(t) = (1/\omega_{0}) sin(\omega_{0}t) for

t > 0.

For t < 0:

Close in upper half-plane. No poles inside. G(t) = 0 for t < 0 (causality

Result:

G_{ret}(t) = \theta(t) sin(\omega_{0}t)/\omega_{0}. The driven oscillator response — zero before the impulse, osc

illates after.

Definition CA.1 — Common Traps

Analytic is stronger than differentiable on a line: complex differentiability constrains all directions.
Poles and branch points differ: only isolated poles have ordinary residues.
Contour orientation matters: reversing direction changes the sign of the integral.
Branch cuts are choices: physical results must be consistent across the chosen branch.

Exercises — CA.1–CA.4 Complex Analysis

Using the residue theorem, evaluate the integral \int_{-}\infty^\infty dx/(1+x^{4}

Straightforward

State the value of the Dirichlet integral \int_{0}^\infty sin(x)/x dx and give its numerical value

Straightforward

Evaluate \int_{0}^\infty sin(x)/x dx = \pi/2 using contour integration. Which contour do you use and wh

Straightforward

Use the Bromwich contour (inverse Laplace transform) to find f(t) = L^(-1

{

1/(s^{2}+\omega^{2}

}. Compare to the Green's function result.

Intermediate

5.Use a conf

ormal map to solve the electrostatic potential between two parallel conducting plates at \phi

. How does this generalize to irregular geometries?

Intermediate

6.Derive Stirling's approximation for n! using the saddle-point method on the Gamma function integral. What is the reflection formula?

Challenging

Key Takeaways

$Cauchy-Riemann: \partialu/\partialx = \partialv/\partialy, \partialu/\partialy = -\partialv/\partialx — analytic ↔ real/imaginary parts harmonic$ .
$Residue theorem: \ointf dz = 2\pii \sum Res — converts integrals to sums of local residues$ .
Real integrals via contour: close in upper or lower half-plane; Jordan's lemma kills the arc.
$Kramers-Kronig: causality forces Re(\chi) and Im(\chi) to be Hilbert transform pairs$ .
$Laurent series: a_{-1} = residue. Pole order from principal part; essential singularity has \infty$ terms.
Conformal maps preserve angles, reduce Laplace to simpler domains — key tool in 2D electrostatics.