Complex Analysis
🟢 Lite — Quick Review (1h–1d)
Core Theorems & Formulas
| Concept | Key Result |
|---|---|
| Cauchy-Riemann (CR) | uₓ = vᵧ, uᵧ = −vₓ for f(z)=u+iv to be analytic |
| Cauchy Integral Theorem | ∮ f(z) dz = 0 if f is analytic in simply connected domain |
| Cauchy Integral Formula | f(z₀) = (1/2πi)∮ f(z)/(z−z₀) dz |
| Laurent Series | f(z) = Σaₙ(z−z₀)ⁿ valid in annular region |
| Residue Theorem | ∮ f(z) dz = 2πi Σ Res(f, poles inside C) |
⚡ Quick记住: Analytic = differentiable everywhere in region (not just at a point). Entire = analytic everywhere.
🟡 Standard — Regular Study (2d–2mo)
Analytic Functions
Definition
f(z) = u(x,y) + iv(x,y) is analytic (holomorphic) at z₀ if f’(z₀) exists — and crucially, this derivative must be the same from ALL directions in the complex plane.
Cauchy-Riemann Equations
Necessary condition: At a point where f is analytic: $$u_x = v_y \quad \text{and} \quad u_y = -v_x$$
Sufficient condition: If uₓ, vᵧ, uᵧ, vₓ exist and are continuous AND CR equations hold in a neighbourhood, then f is analytic.
⚡ GATE Check: Always verify CR first when asked “is f analytic?”
Harmonic Functions
- u and v individually satisfy Laplace’s equation: uₓₓ + uᵧᵧ = 0
- v is the harmonic conjugate of u (or vice versa)
- If v is harmonic conjugate of u: f’(z) = uₓ + ivₓ = vᵧ − iuᵧ
Pai to find harmonic conjugate: Given u(x,y), find v such that vₓ = −uᵧ and vᵧ = uₓ, then integrate and check consistency.
Conformal Mapping
Definition
A mapping w = f(z) is conformal at z₀ if it preserves angles (orientation also preserved).
Key Property
If f is analytic and f’(z₀) ≠ 0, the mapping is conformal at z₀.
- f’(z₀) gives the scale factor (magnification) and rotation angle
- |f’(z₀)| = scaling factor, arg f’(z₀) = rotation
Application: Map problems from z-plane to w-plane where boundaries become simple (e.g., half-plane → unit circle).
Standard Mappings
| Mapping | Effect |
|---|---|
| w = z − a | Translation by a |
| w = e^(iz) | Maps strip 0 < Im z < π to upper half-plane |
| w = (z−a)/(z−b) | Maps circles to lines or circles |
Cauchy Integral Theorem
Statement: If f is analytic in a simply connected domain D, and C is any closed contour in D: $$\oint_C f(z) dz = 0$$
⚡ Key: The domain must be simply connected (no holes). If there’s a pole inside, you can’t apply this directly.
Consequence: The integral is path-independent — only the endpoints matter for analytic f.
Cauchy Integral Formula
Formula: If f is analytic inside and on C, and z₀ is inside C: $$f(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - z_0} dz$$
Extension — Derivatives: $$f^{(n)}(z_0) = \frac{n!}{2\pi i} \oint_C \frac{f(z)}{(z - z_0)^{n+1}} dz$$
This is remarkable: derivatives of all orders are determined by values on the boundary!
Laurent Series
Definition
f(z) = Σaₙ(z−z₀)ⁿ expanded around z₀, valid in annulus r₁ < |z−z₀| < r₂
- Regular part: Σaₙ(z−z₀)ⁿ for n ≥ 0 (Taylor part)
- Principal part: Σaₙ(z−z₀)ⁿ for n < 0 (negative powers)
Classification
- Removable singularity: All negative coefficients vanish (limit exists)
- Pole of order m: Principal part has finitely many terms, highest is m/(z−z₀)^m
- Essential singularity: Infinitely many negative powers (Picard: takes all values except possibly one)
⚡ GATE Test: To classify singularity at z₀, expand in Laurent series or use limit tests.
Residue Theorem
Residue at pole z₀: The coefficient a₋₁ in Laurent expansion = Res(f, z₀)
Evaluation shortcuts:
| Pole type | Residue formula |
|---|---|
| Simple pole (order 1) | lim_(z→z₀) (z−z₀)f(z) |
| Order m pole | (1/(m−1)!) lim_(z→z₀) d^(m−1)/dz^(m−1)[(z−z₀)^m f(z)] |
Residue Theorem: $$\oint_C f(z) dz = 2\pi i \sum_{k} \text{Res}(f, z_k)$$
Sum over all poles inside contour C.
🔴 Extended — Deep Study
CR Equations — Derivation & Examples
For f(z) = u + iv to be differentiable along any path: $$f’(z) = \lim_{\Delta z \to 0} \frac{f(z+\Delta z) - f(z)}{\Delta z}$$
Taking Δz → 0 along real axis: f’(z) = uₓ + ivₓ Taking Δz → 0 along imaginary axis: f’(z) = −ivᵧ + uᵧ
Equating these gives CR: uₓ = vᵧ, uᵧ = −vₓ
Example (GATE 2021): f(z) = z³ + 2z
- u = x³ − 3xy² + 2x, v = 3x²y − y³ + 2y
- uₓ = 3x² − 3y² + 2, vᵧ = 3x² − 3y² + 2 ✓
- uᵧ = −6xy, vₓ = 6xy ✓
- f is entire (analytic everywhere) ✓
Conformal Mapping Applications
Riemann Mapping Theorem: Any non-empty simply connected domain ≠ ℂ can be mapped conformally onto the unit disk. The actual mapping is complicated to find, but existence is guaranteed.
GATE Application: Use w = e^(iz) to map strip to half-plane. Use linear fractional transforms w = (z−a)/(z−b) to map circles/lines.
Key property preserved under conformal maps:
- Angles between curves (conformality)
- Harmony (Laplace’s equation form is invariant under conformal maps — if u solves Laplace in one domain, the mapped function solves Laplace in the image domain)
Cauchy Integral Formula — Worked
Example: Evaluate ∮_|z|=2 (e^z)/(z − πi/2) dz
By CIF: f(z) = e^z is entire, z₀ = πi/2 is inside |z|=2 $$\oint \frac{e^z}{z - \pi i/2} dz = 2\pi i \cdot e^{\pi i/2} = 2\pi i \cdot i = -2\pi$$
(f(z₀) = e^(πi/2) = i)
Laurent Series — Classification
Pole Identification Without Full Series
| Test | Result |
|---|---|
| lim_(z→z₀) f(z) = finite | Removable singularity |
| lim_(z→z₀) (z−z₀)^m f(z) = finite non-zero for some m | Pole of order m |
| lim_(z→z₀) (z−z₀)^m f(z) has no finite limit for any m | Essential singularity |
GATE 2020: e^(1/z) at z=0 — expand: 1 + 1/z + 1/(2!z²) + … — infinite negative powers → essential singularity.
Residue Theorem — GATE Worked Example
Evaluate: ∮_|z|=1 e^(1/z) dz
- z=0 is the only singularity inside |z|=1
- Laurent of e^(1/z): 1 + 1/z + 1/(2!z²) + 1/(3!z³) + …
- Coefficient of 1/z: a₋₁ = 1 → Res = 1
- Integral: 2πi × 1 = 2πi
⚡ Shorter method: For simple pole at z=0, Res = lim_(z→0) z·e^(1/z) — but limit requires expansion or series.
GATE Previous-Year Highlights
| Year | Problem | Key Concept |
|---|---|---|
| 2018 | Find analytic f s.t. u = x² − y² + 2y | CR + harmonic conjugate |
| 2019 | Evaluate ∮ cos z/(z² dz) over unit circle | Simple pole, Residue = 1 |
| 2020 | Show f(z) = |z|² is not analytic anywhere | CR fails everywhere |
| 2021 | Find conformal map: Im z > 0 → |w| < 1 | w = (z−i)/(z+i) or similar |
| 2022 | Laurent series of 1/(z−1)(z−2) in 1<|z|<2 | Principal part from z=2 singularity |
| 2023 | Evaluate ∮ e^z/z dz around z=0 | CIF: result = 2πi |
| 2023 | Classify singularity of sin z/z⁴ at z=0 | Pole of order 3 (check Laurent) |
⚡ GATE Warning: The most common error is forgetting that |z|² = x² + y² is NOT analytic — CR equations fail immediately. Watch for “modulus” or “conjugate” in function definition.
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