Complex Analysis – Complete Reference Guide

Complex Numbers

Algebraic Form

$z = x + iy$ where $i^2 = -1$

$\text{Re}(z) = x$, $\text{Im}(z) = y$

Polar Form

$z = r e^{i\theta}$ where $r = |z|$

$r = \sqrt{x^2 + y^2}$

$\theta = \arg(z) = \arctan(y/x)$

Euler's Formula

$$e^{i\theta} = \cos\theta + i\sin\theta$$

Operations

$z_1 z_2 = r_1 r_2 e^{i(\theta_1+\theta_2)}$

$\bar{z} = r e^{-i\theta} = x - iy$

$|z| = \sqrt{z \bar{z}}$

De Moivre: $z^n = r^n e^{in\theta}$

Complex Function Basics

Definition

$f(z) = u(x,y) + i v(x,y)$

Separates real and imaginary parts

Continuity

$\lim_{z \to z_0} f(z) = f(z_0)$

Requires limit from all directions

Differentiability

$$f'(z_0) = \lim_{h \to 0} \frac{f(z_0+h)-f(z_0)}{h}$$

Must exist independent of direction

This constraint leads to Cauchy-Riemann equations!

Cauchy-Riemann Equations

CR Equations

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$$

$$\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$

Compact Form

$$\frac{\partial f}{\partial \bar{z}} = 0$$

Consequence

If $u, v$ satisfy CR + continuous partials $\implies$ $f$ is analytic

Analytic $\implies$ infinitely differentiable

Derivative

$$f'(z) = u_x + i v_x = v_y - i u_y$$

Failure to satisfy CR means function is nowhere differentiable!

Elementary Functions

Exponential

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

Entire, period $2\pi i$

Logarithm

$$\log z = \log r + i(\theta + 2\pi k)$$

Multi-valued; requires branch cut

Principal: $-\pi < \text{Im}(\log z) \le \pi$

Powers

$$z^a = e^{a \log z}$$

Trig & Hyperbolic

$\cos z = \frac{e^{iz} + e^{-iz}}{2}$

$\sin z = \frac{e^{iz} - e^{-iz}}{2i}$

Contour Integration

Line Integral

$$\int_C f(z) dz = \int_a^b f(z(t)) z'(t) dt$$

Parameterize: $z(t) = x(t) + iy(t)$

Estimation Lemma

$$\left|\int_C f(z)dz\right| \le M \cdot L$$

$M = \max|f|$ on $C$, $L =$ length of $C$

Independence

In simply connected domain: integral depends only on endpoints

Close contours give same result if homotopic

Use semicircles at $\infty$ for real integral tricks!

Cauchy's Theorem

Statement

$$\oint_C f(z) dz = 0$$

$f$ analytic in simply connected $D$

$C$ closed contour in $D$

Implications

Integral of analytic function around closed path = 0

Path independence in analytic domain

General Form

$$\oint_C f(z) dz = \sum_k \text{Res}(f, z_k) \cdot 2\pi i$$

For multiply connected domains, subtract inner contours

Singularities inside $C$ are exception!

Cauchy's Integral Formula

Basic Formula

$$f(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-z_0} dz$$

$z_0$ inside simple closed contour $C$

Derivatives

$$f^{(n)}(z_0) = \frac{n!}{2\pi i} \oint_C \frac{f(z)}{(z-z_0)^{n+1}} dz$$

Analytic = Power Series

Analyticity guaranteed to have Taylor expansion

Series converges in disk around $z_0$

Any analytic function is smooth and has all derivatives!

Taylor Series

Expansion

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

$$a_n = \frac{f^{(n)}(z_0)}{n!}$$

Radius of Convergence

$R = \frac{1}{\limsup |a_n|^{1/n}}$

Converges $|z-z_0| < R$

Key Properties

Unique within disk of convergence

Can differentiate/integrate term-by-term

$R$ determined by distance to nearest singularity!

Laurent Series

Expansion

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

Valid in annulus $r < |z-z_0| < R$

Coefficients

$$a_n = \frac{1}{2\pi i}\oint_C \frac{f(z)}{(z-z_0)^{n+1}}dz$$

Principal Part

$\sum_{n=-\infty}^{-1} a_n(z-z_0)^n$

Describes behavior near singularity

Residue

$$\text{Res}(f, z_0) = a_{-1}$$

Types of Singularities

Removable

$\lim_{z \to z_0} f(z) = L$ exists

Redefine $f(z_0) = L$ $\implies$ analytic

Pole of Order $n$

$f(z) = \frac{g(z)}{(z-z_0)^n}$, $g$ analytic, $g(z_0) \ne 0$

Laurent: $a_{-n} \ne 0$ but $a_k = 0$ for $k < -n$

Essential

Laurent has infinitely many negative powers

Picard theorem: $f$ takes all values near $z_0$

Great Picard: essential singularity misses at most 1 value!

Residue Theory

Residue Theorem

$$\oint_C f(z)dz = 2\pi i \sum_k \text{Res}(f, z_k)$$

Sum over all singularities inside $C$

Computing Residues

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

Pole order $n$: $\text{Res}(f,z_0) = \frac{1}{(n-1)!}\lim \frac{d^{n-1}}{dz^{n-1}}[(z-z_0)^n f(z)]$

Laurent Expansion

Extract $a_{-1}$ coefficient directly

Most practical method: use residue formula for poles!

Argument Principle

Statement

$$\frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)} dz = N - P$$

$N$ = number of zeros inside $C$

$P$ = number of poles inside $C$

Interpretation

Winding number of $f(C)$ around origin

Counts zeros - poles (with multiplicity)

Application

Locate zeros without finding them explicitly

Prove existence of solutions

Uses logarithmic derivative $\frac{f'}{f}$!

Rouché's Theorem

Statement

If $|g(z)| < |f(z)|$ on contour $C$

Then $f$ and $f+g$ have same number of zeros inside $C$

Key Idea

Perturbation doesn't change zero count

Dominating term controls behavior

Example

$p(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_0$

Has $n$ roots (Fundamental Theorem of Algebra)

Compare $z^n$ with lower terms to prove existence!

Maximum Modulus Principle

Statement

If $f$ analytic in domain $D$

Then $|f|$ has no local max in interior of $D$

Consequence

Max of $|f|$ attained on boundary

Unless $f$ is constant

Minimum Modulus

If $f(z) \ne 0$ in $D$: $|f|$ attains min on boundary

Open Mapping

Analytic non-constant maps open sets to open sets

Liouville's Theorem

Statement

Entire bounded function is constant

Consequence

$$\text{If } |f(z)| \le M \text{ for all } z \in \mathbb{C}$$

Then $f(z) = c$ (constant)

Proof Idea

Use Cauchy's integral formula for derivative

Boundedness $\implies$ $f'(z) = 0$ everywhere

App: Fund. Thm Algebra

Non-constant polynomial has at least one zero

Powerful! Shows analytic functions are "rigid"

Conformal Mappings

Definition

Map preserving angles between curves

Analytic $f$ with $f'(z) \ne 0$ is conformal

Properties

Locally preserves shape (angle & orientation)

Infinitesimal circles map to circles

Geometric Mappings

$w = z^2$ doubles angles

$w = e^z$ strips $\to$ annuli

$w = \sin z$ slits $\to$ planes

Transform boundary problems via conformal maps!

Möbius Transformations

General Form

$$f(z) = \frac{az+b}{cz+d}, \quad ad-bc \ne 0$$

Properties

Conformal automorphism of extended plane

3 parameters (normalize $ad-bc=1$)

Fixed Points

Solve $f(z) = z$ for fixed points

Classify: elliptic, parabolic, hyperbolic

Composition

Composition of Möbius = Möbius

Group structure: $PSL(2,\mathbb{C})$

Harmonic Functions

Definition

$$\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$$

Connection to Analytic

If $f = u + iv$ analytic, both $u,v$ harmonic

$u,v$ are harmonic conjugates

Mean Value Property

$$u(z_0) = \frac{1}{2\pi}\int_0^{2\pi} u(z_0 + re^{i\theta})d\theta$$

Max Principle

Harmonic max on boundary of domain

PDEs solved via conformal maps + harmonic functions!

Residue Integral Applications

Type 1: Rational Functions

$$\int_{-\infty}^\infty \frac{P(x)}{Q(x)}dx = 2\pi i \sum \text{Res (upper half)}$$

$\deg(Q) \ge \deg(P) + 2$

Type 2: Trig Integrals

$$\int_0^{2\pi} R(\cos\theta, \sin\theta)d\theta$$

Substitute $z = e^{i\theta}$, close on $|z|=1$

Type 3: Jordan Lemma

$\int_{-\infty}^\infty e^{iax} f(x)dx$ with $a > 0$

Close in upper half-plane

Most powerful technique for real integrals!

Complex Analysis Complete Reference