Complex Analysis - Complex Numbers and Analytic Functions

Table of Contents

1. Complex Numbers
2. Complex Functions
3. Analyticity and Cauchy-Riemann Equations
4. Complex Integration
5. Cauchy’s Theorems
6. Power Series
7. Residue Theory
8. Conformal Mappings

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Complex Numbers

Representation

z = x + iy where x, y ∈ ℝ and i² = -1

Real part: Re(z) = x Imaginary part: Im(z) = y

Polar Form

z = r(cos θ + i sin θ) = re^(iθ)

where r = |z| = √(x² + y²) and θ = arg(z) = arctan(y/x)

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

Complex Operations

Addition: (x₁ + iy₁) + (x₂ + iy₂) = (x₁ + x₂) + i(y₁ + y₂)

Multiplication: (x₁ + iy₁)(x₂ + iy₂) = (x₁x₂ - y₁y₂) + i(x₁y₂ + y₁x₂)

Division: (z₁/z₂) = z₁z̄₂/|z₂|²

Conjugate: z̄ = x - iy

Modulus: |z| = √(zz̄) = √(x² + y²)

Argument: arg(z) (multi-valued, principal value in (-π, π])

De Moivre’s Theorem

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

(e^(iθ))^n = e^(inθ) = cos(nθ) + i sin(nθ)

Roots

nth roots of unity: z^n = 1

Solutions: e^(2πik/n) for k = 0, 1, …, n-1

General nth roots of z: z^(1/n) = |z|^(1/n) e^(i(θ + 2πk)/n) for k = 0, 1, …, n-1

Logarithm

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

Multi-valued: Add 2πik for any integer k

Principal branch: -π < arg(z) ≤ π

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Complex Functions

Domain and Range

Function f: D ⊆ ℂ → ℂ

Domain: Set where function defined Range: Set of function values

Exponential Function

$e^z = e^x (\cos y + i\sin y)$

Properties:

• e^(z₁+z₂) = e^(z₁)e^(z₂)
• |e^z| = e^(Re z)
• e^(z+2πi) = e^z (periodic with period 2πi)
• e^(-z) = 1/e^z

Trigonometric Functions

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

Properties:

• sin²z + cos²z = 1
• sin(z₁+z₂) = sin z₁ cos z₂ + cos z₁ sin z₂
• cos(z₁+z₂) = cos z₁ cos z₂ - sin z₁ sin z₂
• Derivatives: (sin z)’ = cos z, (cos z)’ = -sin z

Hyperbolic Functions

sinh z = (e^z - e^(-z))/2 cosh z = (e^z + e^(-z))/2 tanh z = sinh z/cosh z

Identities:

• cosh²z - sinh²z = 1
• sinh(iz) = i sin z
• cosh(iz) = cos z

Logarithm

Multivalued: log z = ln|z| + i arg(z) = ln r + i(θ + 2πk)

Principal branch: Log z with -π < arg(z) ≤ π

Derivative: d/dz (Log z) = 1/z (away from branch cut)

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Analyticity and Cauchy-Riemann Equations

Limit and Continuity

Limit: $\lim_{z \to z_0} f(z) = L$ if for ε > 0, there exists δ > 0 such that |z - z_0| < δ ⟹ |f(z) - L| < ε

Continuity: f continuous at z₀ if limit exists and equals f(z₀)

Differentiability

Derivative: $f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0}$

Must exist and be same regardless of approach direction.

Cauchy-Riemann Equations

Write z = x + iy, f(z) = u(x,y) + iv(x,y)

Necessary conditions for differentiability:

If f’(z) exists, then: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

Sufficient conditions:

If u,v have continuous first partial derivatives and satisfy C-R equations, then f is analytic.

Analytic Functions

Analytic (holomorphic) at point: f differentiable in neighborhood of point

Entire: Analytic everywhere in ℂ

Constant: If f analytic and f’ = 0 on connected domain, then f constant.

Harmonic Functions

If f = u + iv is analytic, then u and v are harmonic.

Harmonic: ∇²u = u_{xx} + u_{yy} = 0

u and v are harmonic conjugates.

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Complex Integration

Contour (Path)

Parameterization: γ(t) = x(t) + iy(t), a ≤ t ≤ b

Closed curve: γ(a) = γ(b)

Simple curve: No self-intersections

Positively oriented: Counter-clockwise (usually)

Integral Along Contour

$\int_{\gamma} f(z)dz = \int_a^b f(\gamma(t))\gamma'(t)dt$

Properties:

• ∫(f + g)dz = ∫f dz + ∫g dz
• ∫cf dz = c∫f dz
• ∫[-γ] f dz = -∫[γ] f dz (reverse orientation)
• ∫[γ₁+γ₂] f dz = ∫[γ₁] f dz + ∫[γ₂] f dz

Estimation Lemma

If |f(z)| ≤ M on curve γ with length L, then: $|\int_{\gamma} f(z)dz| \leq ML$

Application: Jordan’s lemma for semicircles.

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Cauchy’s Theorems

Cauchy-Goursat Theorem

If f is analytic on simply connected domain D, then for any closed contour γ in D:

$\int_{\gamma} f(z)dz = 0$

Simply connected: Every closed curve can be contracted to point.

Cauchy Integral Formula

If f analytic on/inside simple closed contour γ, and z inside γ:

$f(z) = \frac{1}{2\pi i}\int_{\gamma} \frac{f(\xi)}{\xi - z}d\xi$

Differentiation under integral sign:

$f^{(n)}(z) = \frac{n!}{2\pi i}\int_{\gamma} \frac{f(\xi)}{(\xi-z)^{n+1}}d\xi$

Liouville’s Theorem

Bounded entire function is constant.

Consequence: Fundamental Theorem of Algebra (polynomial has at least one root in ℂ)

Maximum Modulus Principle

If f analytic on bounded domain, |f| attains maximum on boundary.

Consequence: If |f| constant on boundary and f analytic, then f is constant.

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Power Series

Taylor Series

If f analytic in disk |z - z₀| < R:

$f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n$

Radius of convergence: Distance to nearest singularity.

Laurent Series

f analytic in annulus r < |z - z₀| < R:

$f(z) = \sum_{n=-\infty}^{\infty} a_n(z-z_0)^n$

where: $a_n = \frac{1}{2\pi i}\int_{\gamma} \frac{f(\xi)}{(\xi-z_0)^{n+1}}d\xi$

Principal part: Negative powers Analytic part: Nonnegative powers

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Residue Theory

Isolated Singularities

Removable: Limit exists (can redefine) Pole of order m: (z-z₀)^m f(z) bounded and nonzero Essential: Neither removable nor pole

Residue

At pole z₀ of order m:

$\text{Res}(f, z_0) = \frac{1}{(m-1)!}\lim_{z \to z_0}\frac{d^{m-1}}{dz^{m-1}}[(z-z_0)^m f(z)]$

Simple pole (m=1):

$\text{Res}(f, z_0) = \lim_{z \to z_0}(z-z_0)f(z)$

If f = g/h with g(z₀) ≠ 0, h(z₀) = 0, h’(z₀) ≠ 0:

$\text{Res}(f, z_0) = \frac{g(z_0)}{h'(z_0)}$

Residue Theorem

If f analytic inside and on simple closed contour γ except for isolated singularities:

$\int_{\gamma} f(z)dz = 2\pi i \sum \text{Res}(f, z_k)$

Sum over all singularities inside γ.

Applications to Real Integrals

Rational functions:

Evaluate ∫∞_{-∞} P(x)/Q(x) dx where deg Q ≥ deg P + 2

Close in upper half-plane, sum residues.

Fourier integrals:

Evaluate ∫∞_{-∞} f(x)e^(iax) dx (a > 0)

Close in upper half-plane.

Integrals with trig functions:

Substitute z = e^(iθ) to convert to contour integral around unit circle.

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Conformal Mappings

Conformality

Map preserves angles if f’(z) ≠ 0

Orientation preserving if |arg f’(z)| < π

Analytic function with nonzero derivative is conformal.

Möbius Transformations

Form: f(z) = (az + b)/(cz + d) with ad - bc ≠ 0

Properties:

• Maps circles/lines to circles/lines
• Preserves cross-ratio
• Three points determine transformation

Common forms:

• Translation: f(z) = z + b
• Rotation: f(z) = e^(iθ) z
• Dilation: f(z) = kz (k > 0)
• Inversion: f(z) = 1/z

Schwarz-Christoffel Transformation

Maps upper half-plane to polygon interior.

Formula: f’(z) = AΠ(z - z_k)^(α_k/π - 1)

where α_k are interior angles.

Joukowski Transformation

$f(z) = z + 1/z$

Maps circles to airfoil shapes (applications in aerodynamics).

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Next: Discrete Mathematics or Differential Equations

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Last updated: Comprehensive complex analysis reference covering analytic functions, integration, and residue theory.