Prerequisite:

The Complex Numbers

The complex numbers $\mathbb{C} = {x + iy : x, y \in \mathbb{R}}$ extend $\mathbb{R}$ by adjoining $i$ with $i^2 = -1$. Every $z = x + iy$ has:

  • Real part $\operatorname{Re}(z) = x$ and imaginary part $\operatorname{Im}(z) = y$.
  • Modulus $|z| = \sqrt{x^2 + y^2}$ (distance from origin).
  • Argument $\arg(z) = \theta$ where $z = |z|e^{i\theta}$ (angle from positive real axis).
  • Complex conjugate $\bar{z} = x - iy$, with $z\bar{z} = |z|^2$.

Euler’s Formula. For any $\theta \in \mathbb{R}$:

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

This follows from the Taylor series of $e^z$, $\cos\theta$, and $\sin\theta$. Every complex number can be written in polar form $z = r e^{i\theta}$ where $r = |z|$ and $\theta = \arg(z)$. Multiplication becomes $z_1 z_2 = r_1 r_2,e^{i(\theta_1 + \theta_2)}$ - multiply moduli, add arguments.

        Im
         |
         |     . z = x + iy = r*e^{i*theta}
         |    /|
       r |   / |
         |  /  | y
         | /   |
         |/ theta
---------+-----------> Re
         |   x
         |
The complex plane: Re axis horizontal, Im axis vertical.

De Moivre’s Theorem. $(e^{i\theta})^n = e^{in\theta}$, so $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$.

Complex Functions

A complex function $f: \mathbb{C} \to \mathbb{C}$ can be written as $f(z) = u(x,y) + iv(x,y)$ where $u$ and $v$ are real-valued functions of two real variables. For example, $f(z) = z^2 = (x^2 - y^2) + i(2xy)$, so $u = x^2 - y^2$ and $v = 2xy$.

Holomorphic Functions

Definition. A function $f$ is complex differentiable at $z_0$ if the limit

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

exists, where $h \in \mathbb{C}$ and the limit is the same regardless of the direction in which $h \to 0$. A function that is complex differentiable at every point of an open set $U$ is called holomorphic on $U$.

The direction-independence is a strong condition - far stronger than real differentiability. It forces the Cauchy-Riemann equations.

Cauchy-Riemann Equations

Theorem (Cauchy-Riemann). If $f = u + iv$ is holomorphic at $z_0 = x_0 + iy_0$, then the partial derivatives of $u$ and $v$ satisfy

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

at $(x_0, y_0)$. Conversely, if $u$ and $v$ have continuous partial derivatives satisfying the CR equations at a point, then $f$ is holomorphic there.

Proof sketch. Consider the limit $h \to 0$ along the real axis ($h = \Delta x$) and along the imaginary axis ($h = i\Delta y$). Setting the two results equal and separating real/imaginary parts gives the CR equations.

Consequence. Both $u$ and $v$ are harmonic: $\nabla^2 u = u_{xx} + u_{yy} = 0$ and $\nabla^2 v = 0$. This follows by differentiating the CR equations.

Types of Singularities

A point $z_0$ is an isolated singularity of $f$ if $f$ is holomorphic in a punctured disk $0 < |z - z_0| < r$ but not at $z_0$ itself.

  • Removable singularity: $\lim_{z \to z_0} f(z)$ exists (e.g., $\sin(z)/z$ at $z=0$).
  • Pole of order $m$: $|f(z)| \to \infty$ as $z \to z_0$, and $(z - z_0)^m f(z)$ has a removable singularity.
  • Essential singularity: neither of the above (e.g., $e^{1/z}$ at $z = 0$ - by Picard’s theorem, takes every complex value except possibly one in any punctured neighborhood).

Entire functions (holomorphic on all of $\mathbb{C}$) include $e^z$, $\sin z$, $\cos z$, and all polynomials.

Complex Integration

For a smooth curve $\gamma: [a,b] \to \mathbb{C}$, the contour integral is

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

Unlike real integration, the result depends on the path - unless $f$ is holomorphic.

Cauchy’s Integral Theorem

Theorem (Cauchy). If $f$ is holomorphic on a simply connected domain $D$, and $\gamma$ is any closed curve in $D$, then

$$\oint_\gamma f(z),dz = 0.$$

“Simply connected” means no holes - any closed curve can be continuously shrunk to a point within $D$. The theorem fails if $D$ has holes: $\oint_{|z|=1} \frac{1}{z},dz = 2\pi i \neq 0$.

Cauchy’s Integral Formula

Theorem. If $f$ is holomorphic on and inside a simple closed curve $\gamma$ (traversed counterclockwise), and $z_0$ is inside $\gamma$, then

$$f(z_0) = \frac{1}{2\pi i}\oint_\gamma \frac{f(z)}{z - z_0},dz.$$

This is remarkable: the value of a holomorphic function at any interior point is completely determined by its values on the boundary. By differentiating under the integral sign, the $n$-th derivative formula follows:

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

In particular, every holomorphic function is infinitely differentiable - a far stronger conclusion than in real analysis.

Laurent Series and Residues

Near an isolated singularity $z_0$, a holomorphic function has a Laurent series expansion:

$$f(z) = \sum_{n=-\infty}^{\infty} a_n,(z - z_0)^n = \cdots + \frac{a_{-2}}{(z-z_0)^2} + \frac{a_{-1}}{z-z_0} + a_0 + a_1(z-z_0) + \cdots$$

The residue of $f$ at $z_0$ is the coefficient $a_{-1}$ of $(z - z_0)^{-1}$:

$$\operatorname{Res}(f, z_0) = a_{-1} = \frac{1}{2\pi i}\oint_{|z-z_0|=\epsilon} f(z),dz.$$

For a simple pole (order 1): $\operatorname{Res}(f, z_0) = \lim_{z \to z_0}(z - z_0)f(z)$.

For a pole of order $m$: $\operatorname{Res}(f, z_0) = \frac{1}{(m-1)!}\lim_{z \to z_0}\frac{d^{m-1}}{dz^{m-1}}\left[(z-z_0)^m f(z)\right]$.

The Residue Theorem

Theorem (Residue Theorem). Let $f$ be holomorphic on a simply connected domain $D$ except at finitely many isolated singularities $z_1, \ldots, z_n$ inside a simple closed curve $\gamma$. Then

$$\oint_\gamma f(z),dz = 2\pi i \sum_{k=1}^n \operatorname{Res}(f, z_k).$$

Cauchy’s integral theorem is the special case with no singularities inside $\gamma$.

Application: Evaluating Real Integrals

The residue theorem lets us compute real integrals that resist elementary methods.

Example. Compute $\displaystyle I = \int_{-\infty}^\infty \frac{1}{1 + x^2},dx$.

Consider $f(z) = \frac{1}{1 + z^2} = \frac{1}{(z+i)(z-i)}$ and the semicircular contour $\Gamma_R$ consisting of $[-R, R]$ on the real axis plus the upper semicircle $|z| = R$, $\operatorname{Im}(z) \geq 0$.

        Im
         |
         |    .-~~~~~-.
         |   /         \    Semicircle C_R
         |  |     +i    |   (radius R -> inf)
         |  |  (pole)   |
---------+--+-----+-----+--> Re
        -R         R
Real axis segment [-R, R]

The only pole inside $\Gamma_R$ is $z = i$ (since $z = -i$ is in the lower half-plane). The residue is $\operatorname{Res}(f, i) = \frac{1}{z + i}\big|_{z=i} = \frac{1}{2i}$.

By the residue theorem: $\oint_{\Gamma_R} f,dz = 2\pi i \cdot \frac{1}{2i} = \pi$.

The semicircle contribution vanishes as $R \to \infty$ (Jordan’s lemma), leaving $I = \pi$.

Conformal Mappings and Möbius Transformations

A holomorphic function $f$ with $f'(z_0) \neq 0$ is conformal at $z_0$: it preserves angles between curves. This makes conformal maps powerful for solving boundary-value problems - map a complicated domain to a simple one (e.g., the unit disk), solve there, map back.

Möbius transformations (or fractional linear transformations) have the form

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

They map circles and lines to circles and lines, preserve angles, and form a group under composition. The Riemann mapping theorem guarantees that any simply connected proper subset of $\mathbb{C}$ is conformally equivalent to the unit disk.

Liouville’s Theorem and the Fundamental Theorem of Algebra

Theorem (Liouville). Every bounded entire function is constant.

Proof. If $|f(z)| \leq M$ for all $z$ and $f$ is entire, Cauchy’s integral formula gives $|f'(z_0)| \leq M/R$ for any $R > 0$. Letting $R \to \infty$ gives $f'(z_0) = 0$ for all $z_0$.

Theorem (Fundamental Theorem of Algebra). Every non-constant polynomial $p(z) \in \mathbb{C}[z]$ has at least one root in $\mathbb{C}$.

Proof via Liouville. If $p(z) \neq 0$ for all $z$, then $f(z) = 1/p(z)$ is entire. Since $|p(z)| \to \infty$ as $|z| \to \infty$, we have $f(z) \to 0$, so $f$ is bounded. By Liouville, $f$ is constant, contradicting that $p$ is non-constant.

Examples

Z-Transform ROC. The Z-transform $X(z) = \sum_n x[n]z^{-n}$ is the discrete analogue of the Laplace transform. The ROC is an annulus in the complex $z$-plane. Stability of a discrete-time system requires all poles inside the unit circle $|z| < 1$ - the discrete counterpart of the left half-plane condition.

Signal Processing. The DFT evaluates the Z-transform on the unit circle: $X(e^{i\omega}) = \sum_n x[n]e^{-i\omega n}$. Poles near the unit circle cause peaks in the frequency response - the mechanism behind resonant filters.

Quantum Computing. Quantum states are unit vectors in $\mathbb{C}^n$; amplitudes are complex numbers. The Born rule gives probability $|a|^2$ for amplitude $a \in \mathbb{C}$. Quantum gates are unitary matrices over $\mathbb{C}$. The interference of complex amplitudes - destructive and constructive - is what makes quantum algorithms (e.g., Grover’s, Shor’s) more powerful than classical ones.

Complex Gradients. Wirtinger calculus extends complex differentiation to non-holomorphic functions (like $|z|^2$) by treating $z$ and $\bar{z}$ as independent variables: $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \bar{z}}$. This is used in complex-valued neural networks and in the derivation of natural gradient methods.

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