Complex Analysis

Complex Analysis

🟢 Lite — Quick Review (1h–1d)

Complex Analysis studies functions of a complex variable f(z) where z = x + iy, primarily in the context of analyticity, contour integration, and singularities in the complex plane.

Key definitions:

• Analytic: f(z) is differentiable at every point in a neighborhood — not just at a point.
• Cauchy-Riemann (C-R) equations: for f(z) = u(x,y) + iv(x,y), analyticity requires uₓ = vᵧ and uᵧ = −vₓ.
• Singularity types: removable (limit finite), pole (order n finite), essential (infinite order).
• Residue at z = a: coefficient of 1/(z−a) in Laurent series.

Must-know formulas:

• C-R: uₓ = vᵧ, uᵧ = −vₓ
• Cauchy integral formula: f(a) = (1/2πi) ∮ f(z)/(z−a) dz
• Residue theorem: ∮ f(z) dz = 2πi × (sum of residues inside C)

High-yield exam pointers for GATE:

1. GATE Engineering Maths carries ~13–15 marks total; Complex Analysis typically yields 2–4 marks as 1-mark MCQ or 2-mark NAT.
2. Questions often ask to verify analyticity via C-R equations first — never skip this step.
3. For pole of order n at z = a, use: Res = (1/(n−1)!) × lim_{z→a} d^{n−1}/dz^{n−1}[(z−a)*^n f(z)].
4. Common trap: applying Cauchy’s theorem (integral = 0) when singularities lie inside the contour.

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🟡 Standard — Regular Study (2d–2mo)

Analyticity and the Cauchy-Riemann Equations

A function f(z) = u(x,y) + iv(x,y) is analytic (holomorphic) on a domain D if it is complex-differentiable at every point of D. Differentiability demands that the C-R equations hold:

$$u_x = v_y \quad \text{and} \quad u_y = -v_x$$

and additionally that these partial derivatives are continuous in a neighborhood. Both conditions are required — satisfying C-R alone is insufficient for analyticity unless continuity is confirmed.

Singularities and Their Classification

Let f have an isolated singularity at z₀:

• Removable singularity: lim_{z→z₀} f(z) exists finitely. The function can be redefined to be analytic at z₀.
• Pole of order m: |f(z)| → ∞ as z → z₀, and (z−z₀)^m f(z) is analytic and non-zero at z₀.
• Essential singularity: Laurent series has infinitely many negative powers; behaviour is wild (Picard’s theorem).

Residue Calculation

The residue at a pole of order n at z = a is:

$$\text{Res}{z=a} f(z) = \frac{1}{(n-1)!} \lim{z\to a} \frac{d^{n-1}}{dz^{n-1}}\left[(z-a)^n f(z)\right]$$

For a simple pole (n = 1): Res = lim_{z→a} (z−a) f(z).

Cauchy’s Integral Theorem and Formulae

• Cauchy’s theorem: if f is analytic on and inside a closed contour C, then ∮_C f(z) dz = 0.
• Cauchy integral formula (for derivatives): if f is analytic inside C, then

$$f^{(n)}(a) = \frac{n!}{2\pi i} \oint_C \frac{f(z)}{(z-a)^{n+1}} dz$$

This extends to evaluating higher-order derivatives directly via contour integration.

Residue Theorem for Contour Integration

For a function analytic except at isolated singularities inside C:

$$\oint_C f(z), dz = 2\pi i \sum \text{Residues inside } C$$

GATE typically asks to evaluate real integrals (e.g., ∫₀^{2π} R(sinθ, cosθ) dθ) by converting to a contour integral on |z| = 1.

Concept	Condition	Key formula
Analyticity	C-R + continuity	f′ exists on neighborhood
Simple pole residue	order 1	lim_{z→a} (z−a) f(z)
nth order pole residue	order n	derivative formula above
Cauchy theorem	analytic on & inside C	∮ f dz = 0
Residue theorem	isolated singularities	2πi × sum of residues

Typical GATE question patterns: (a) verify analyticity by applying C-R to a given f, (b) identify singularity type and compute residue, (c) evaluate a closed contour integral using the residue theorem.

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🔴 Extended — Deep Study (3mo+)

Laurent Series and the Ring of Convergence

When singularities lie inside a region, Taylor series fails — the correct tool is the Laurent series:

$$f(z) = \sum_{n=-\infty}^{\infty} a_n (z-a)^n, \quad a_n = \frac{1}{2\pi i} \oint_C \frac{f(\zeta)}{(\zeta-a)^{n+1}} d\zeta$$

The series splits into the analytic part (non-negative powers) and the principal part (negative powers). The coefficient a_{−1} equals the residue. Convergence holds in the annulus R₁ < |z−a| < R₂ where R₁ is distance to the nearest singularity and R₂ is distance to the next one.

Essential Singularities — Weierstrass and Picard

An essential singularity has infinitely many negative terms in its Laurent expansion. Weierstrass’s theorem: near an essential singularity, f(z) gets arbitrarily close to any complex value infinitely often. Picard’s stronger result: f takes every complex value (with at most one exception) infinitely often in any neighborhood of an essential singularity. Example: e^{1/z} has an essential singularity at z = 0.

Singularity at Infinity

Treat w = 1/z; expand around w = 0. Classify the singularity at infinity via the behavior of f(w) at w = 0:

• If f(z) → 0 as |z| → ∞ → removable singularity at infinity.
• If f(z) → constant ≠ 0 → simple pole at infinity.
• If f(z) grows as z^n → pole of order n at infinity.
• If behavior is unbounded in an essential way → essential singularity at infinity.

Residue at infinity: Res_{z=∞} f(z) = −Res_{w=0} (1/w²) f(1/w). Useful when it’s easier to compute the residue at z = ∞ than sum residues elsewhere.

Conformal Mapping

A map w = f(z) is conformal (angle-preserving) if f is analytic with f′(z) ≠ 0. Conformal maps preserve:

• Oriented angles between curves.
• The shape of infinitesimal figures (locally).

Applications in engineering: solving Laplace’s equation (heat transfer, fluid flow) by mapping complicated domains to simpler ones (e.g., unit disk) where boundary conditions are easier to apply.

Common Mistakes and Traps in GATE

1. Assuming Cauchy’s theorem applies when singularities are inside C. If any singularity lies on or inside the contour, the integral is not zero — use the residue theorem instead.
2. Using the wrong order for the residue formula — applying the simple-pole formula to a higher-order pole yields a wrong answer.
3. Skipping the analyticity check before applying Cauchy’s integral formula — the formula requires analyticity on and inside C.
4. Forgetting the singularity at infinity when evaluating integrals over all singularities. The sum of all residues including at infinity equals zero, so Res_{∞} = −∑ all finite residues.
5. Neglecting continuity of C-R partial derivatives — a function can satisfy C-R at a point but still fail to be analytic because the partials are discontinuous there.

Practice Prompts

1. Classify the singularity of f(z) = e^{z}/(z−1)³ at z = 1 and compute Res_{z=1} f(z).

2. Evaluate ∮_C (sin z)/(z(2z−1)) dz where C is |z| = 2, traversed counterclockwise.

3. Use the residue theorem to evaluate the real integral ∫₀^{2π} dθ/(1 + a sin θ) for |a| < 1.

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