Rai Notes

Complex Analysis

8 min read

Introduction

Complex analysis is the study of functions of complex variables. It is remarkably elegant—many results are simpler and more powerful than their real counterparts. Where real analysis often requires careful handling of edge cases, complex analysis rewards us with theorems of surprising generality and beauty.

The subject emerged in the 18th and 19th centuries through the work of Euler, Cauchy, Riemann, and Weierstrass. Today it remains essential in pure mathematics, physics, and engineering.

Complex Numbers

Definition

A complex number has the form:

z = a + bi

where:

  • a is the real part: Re(z) = a
  • b is the imaginary part: Im(z) = b
  • i is the imaginary unit satisfying i² = -1

The complex conjugate is z̄ = a - bi.

The modulus (or absolute value) is:
|z| = √(a² + b²)

Key properties:

  • z · z̄ = |z|²
  • |z₁z₂| = |z₁||z₂|
  • |z₁ + z₂| ≤ |z₁| + |z₂| (triangle inequality)

Polar Form

Any complex number can be written in polar form:

z = r(cos θ + i sin θ) = re^(iθ)

where:

  • r = |z| is the modulus
  • θ = arg(z) is the argument (angle from positive real axis)

Euler’s formula connects exponentials and trigonometry:
e^(iθ) = cos θ + i sin θ

This gives the famous identity: e^(iπ) + 1 = 0

Multiplication in polar form:
z₁z₂ = r₁r₂ · e^(i(θ₁ + θ₂))

Multiply moduli, add arguments.

Division:
z₁/z₂ = (r₁/r₂) · e^(i(θ₁ - θ₂))

The Complex Plane

The Argand diagram represents complex numbers as points in a 2D plane:

  • Horizontal axis: real part
  • Vertical axis: imaginary part

Geometric interpretations:

  • Addition: vector addition
  • Multiplication by e^(iθ): rotation by angle θ
  • Multiplication by r: scaling by factor r
  • Conjugation: reflection across the real axis

The argument arg(z) is the angle θ, typically taken in (-π, π] or [0, 2π).

De Moivre’s Theorem

For any integer n:

(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)

This follows directly from Euler’s formula: (e^(iθ))^n = e^(inθ).

Finding nth roots:
The equation z^n = w has exactly n solutions:

z_k = |w|^(1/n) · e^(i(θ + 2πk)/n) for k = 0, 1, …, n-1

These roots form a regular n-gon in the complex plane.

Complex Functions

Elementary Functions

Polynomials: p(z) = aₙzⁿ + … + a₁z + a₀

Exponential function:
e^z = e^(x+iy) = e^x(cos y + i sin y)

Key property: e^z is periodic with period 2πi.

Trigonometric functions (defined via exponentials):

  • sin z = (e^(iz) - e^(-iz)) / 2i
  • cos z = (e^(iz) + e^(-iz)) / 2

Unlike real trig functions, these are unbounded!

Complex logarithm:
log z = ln|z| + i·arg(z)

This is multi-valued because arg(z) is determined only up to multiples of 2π. The principal branch uses arg(z) ∈ (-π, π].

Complex powers:
z^α = e^(α·log z)

Also multi-valued when α is not an integer.

Limits and Continuity

Limits work similarly to real analysis, but the variable z can approach z₀ along any path in the complex plane.

f(z) → L as z → z₀ means |f(z) - L| → 0 for any approach.

A function is continuous if lim(z→z₀) f(z) = f(z₀).

Analytic Functions

Differentiability

The complex derivative is defined as:

f’(z₀) = lim(z→z₀) [f(z) - f(z₀)] / (z - z₀)

This looks like the real definition, but it is far more restrictive. The limit must exist and be the same regardless of how z approaches z₀—from any direction in the plane.

A function differentiable at every point in a region is called holomorphic (or analytic) there.

Cauchy-Riemann Equations

Write f(z) = u(x,y) + iv(x,y) where u and v are real functions.

If f is analytic, then:

  • ∂u/∂x = ∂v/∂y
  • ∂u/∂y = -∂v/∂x

Conversely, if u and v have continuous partial derivatives satisfying these equations, then f is analytic.

Example: For f(z) = z² = (x² - y²) + i(2xy):

  • u = x² - y², v = 2xy
  • ∂u/∂x = 2x = ∂v/∂y ✓
  • ∂u/∂y = -2y = -∂v/∂x ✓

Harmonic Functions

A real function u(x,y) is harmonic if it satisfies Laplace’s equation:

∇²u = ∂²u/∂x² + ∂²u/∂y² = 0

Key theorem: If f = u + iv is analytic, then both u and v are harmonic.

Given a harmonic function u, we can (locally) find a harmonic conjugate v such that u + iv is analytic.

Analytic vs Holomorphic

In real analysis, differentiable functions need not equal their Taylor series. In complex analysis:

  • Holomorphic: complex differentiable in a region
  • Analytic: equals its power series in a neighbourhood

Remarkable fact: These are equivalent for complex functions!

A single complex derivative implies infinite differentiability and convergent power series representation. This is one of the magical aspects of complex analysis.

Contour Integration

Path Integrals

For a curve C parameterised by z(t), a ≤ t ≤ b:

∫_C f(z) dz = ∫_a^b f(z(t)) · z’(t) dt

The value generally depends on the path taken.

Cauchy’s Theorem

If f is analytic on and inside a simple closed contour C:

∮_C f(z) dz = 0

This is the foundation of complex analysis. Analytic functions have path-independent integrals.

More generally, if f is analytic in a simply connected domain, any closed contour integral vanishes.

Cauchy’s Integral Formula

If f is analytic inside and on a simple closed contour C, and z₀ is inside C:

f(z₀) = (1/2πi) ∮_C f(z)/(z - z₀) dz

The value of an analytic function at any interior point is determined by its values on the boundary!

Derivatives:
f^(n)(z₀) = (n!/2πi) ∮_C f(z)/(z - z₀)^(n+1) dz

This proves analytic functions are infinitely differentiable.

Series Representations

Taylor Series

Every analytic function equals its Taylor series:

f(z) = Σ(n=0 to ∞) [f^(n)(z₀)/n!] · (z - z₀)

The series converges in the largest disc centred at z₀ containing no singularities.

The radius of convergence R satisfies:
1/R = lim sup |aₙ|^(1/n)

Laurent Series

For functions with singularities, we use Laurent series:

f(z) = Σ(n=-∞ to ∞) aₙ(z - z₀)

The principal part consists of negative power terms. It characterises the singularity.

Singularities and Residues

Types of Singularities

An isolated singularity at z₀ means f is analytic near z₀ but not at z₀.

Removable singularity: The limit lim(z→z₀) f(z) exists. The singularity can be “filled in.”
Example: sin(z)/z at z = 0.

Pole of order n: f(z) = g(z)/(z - z₀)^n where g is analytic with g(z₀) ≠ 0.
Example: 1/(z - 1)² has a pole of order 2 at z = 1.

Essential singularity: Neither removable nor a pole. The Laurent series has infinitely many negative powers.
Example: e^(1/z) at z = 0.

Residues

The residue of f at z₀ is the coefficient a₋₁ in the Laurent expansion:

Res(f, z₀) = a₋₁

Computing residues:

  • Simple pole: Res(f, z₀) = lim(z→z₀) (z - z₀)f(z)
  • Pole of order n: Res(f, z₀) = (1/(n-1)!) · lim(z→z₀) d^(n-1)/dz^(n-1) [(z - z₀)^n f(z)]

Residue Theorem

If f is analytic except for isolated singularities inside a closed contour C:

∮_C f(z) dz = 2πi × Σ Res(f, zₖ)

where the sum is over all singularities zₖ inside C.

Applications

Evaluating real integrals: Many difficult real integrals become easy with residue calculus.

Example: ∫_{-∞}^{∞} 1/(1+x²) dx = π (close the contour in the upper half-plane).

Summing series: Residue methods can evaluate sums like Σ 1/n².

Special Results

Liouville’s Theorem

A bounded entire function (analytic on all of ℂ) is constant.

This simple statement has profound consequences.

Fundamental Theorem of Algebra

Every non-constant polynomial has at least one root in ℂ.

Proof sketch: If p(z) had no roots, then 1/p(z) would be entire and bounded (since |p(z)| → ∞ as |z| → ∞). By Liouville’s theorem, 1/p(z) is constant—contradiction.

Maximum Modulus Principle

If f is analytic and non-constant on a domain D, then |f(z)| has no maximum in the interior of D.

The maximum of |f| on a closed bounded region occurs on the boundary.

Applications

Fluid dynamics: The velocity potential and stream function of 2D incompressible flow satisfy the Cauchy-Riemann equations. Conformal mappings transform flow problems.

Electrical engineering: AC circuit analysis uses complex impedance. The phasor representation relies on e^(iωt).

Quantum mechanics: The Schrödinger equation involves complex wave functions. Residue calculus evaluates integrals in scattering theory.

Number theory: The Riemann zeta function ζ(s) is a function of a complex variable. Its properties encode deep information about prime numbers.

Learning Resources

Books

  • Complex Analysis by Joseph Bak and Donald Newman — rigorous yet accessible
  • Visual Complex Analysis by Tristan Needham — highly visual, geometric intuition
  • Complex Analysis: A First Course by Zill and Shanahan — good for beginners
  • Functions of One Complex Variable by John Conway — comprehensive graduate text

Courses

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