9.7 Complex analysis

Fundamentals of complex numbers

Complex numbers extend the real number line by introducing a new dimension. They're the foundation of complex analysis, which turns out to have surprisingly powerful applications across mathematics, physics, and engineering.

Real and imaginary parts

A complex number has two components: a real part and an imaginary part. You write it as:

$z = a + bi$

Here, a is the real part and b is the imaginary part. The key ingredient is the imaginary unit $i$, defined by the property $i^2 = -1$. There's no real number that squares to give $-1$, so $i$ is genuinely new.

Arithmetic works by treating $i$ as a variable and applying $i^2 = -1$ whenever it appears. For example, multiplying $(2 + 3i)(1 - i)$: you expand using FOIL, get $2 - 2i + 3i - 3i^2$, replace $i^2$ with $-1$, and simplify to $5 + i$.

Argand diagram representation

An Argand diagram plots complex numbers on a two-dimensional plane. The horizontal axis is the real part, and the vertical axis is the imaginary part. So $z = 3 + 2i$ gets plotted at the point $(3, 2)$.

• The magnitude (or modulus) of $z$ is the distance from the origin: $|z| = \sqrt{a^2 + b^2}$
• Addition of complex numbers corresponds to vector addition on the diagram
• Multiplication involves both scaling and rotation, which becomes much clearer in polar form

This geometric picture is not just a convenience. It's central to how complex analysis works.

Polar form of complex numbers

Instead of specifying a complex number by its real and imaginary parts, you can use its distance from the origin and its angle:

$z = r(\cos\theta + i\sin\theta)$

Here, $r = |z|$ is the magnitude and $\theta$ (called the argument) is the angle measured counterclockwise from the positive real axis.

Euler's formula connects this to the exponential function:

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

This lets you write the polar form compactly as $z = re^{i\theta}$. The payoff: multiplication becomes simple. To multiply two complex numbers, you multiply their magnitudes and add their angles. Division works by dividing magnitudes and subtracting angles.

Complex functions

A complex function takes a complex number as input and produces a complex number as output. Since both input and output live in two dimensions, you're effectively mapping one plane to another. This richer structure leads to properties that have no analogue in real analysis.

Domain and range

The domain of a complex function is the set of complex numbers where the function is defined. The range is the set of all output values.

• Some functions are defined everywhere (polynomials, $e^z$)
• Others have restrictions: the complex logarithm isn't defined at $z = 0$, and it's actually multi-valued (more on that in the Riemann surfaces section)
• Domain coloring is a visualization technique that assigns colors to output values, letting you "see" how a complex function behaves across its domain

Analytic vs. non-analytic functions

An analytic function is one that is complex-differentiable at every point in its domain and can be expressed as a convergent power series. Analytic functions satisfy the Cauchy-Riemann equations (below) and are infinitely differentiable.

Non-analytic functions fail to be complex-differentiable. Two common examples:

• The complex conjugate $f(z) = \bar{z}$
• The absolute value $f(z) = |z|$

Both of these are perfectly well-defined, but neither is differentiable in the complex sense. The distinction matters because analytic functions have far stronger properties: power series representations, path-independent integrals, and more.

Cauchy-Riemann equations

These equations are the test for complex differentiability. Write a complex function as:

$f(x + iy) = u(x,y) + iv(x,y)$

where $u$ and $v$ are real-valued functions. The Cauchy-Riemann equations are:

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

Why do these matter? Complex differentiability requires the derivative to be the same regardless of which direction you approach from in the complex plane. These two equations are exactly the conditions that guarantee direction-independence. If a function satisfies them (and the partial derivatives are continuous), the function is analytic.

Differentiation in the complex plane

Complex differentiation looks like real differentiation on the surface, but it's actually much more restrictive. A function that's complex-differentiable automatically has far stronger properties than a merely real-differentiable function.

Complex derivatives

The definition mirrors the real case:

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

The critical difference: $h$ is a complex number, so it can approach zero from any direction in the plane. For the limit to exist, it must give the same value no matter how $h \to 0$. This is a much stronger requirement than in real analysis, where $h$ can only approach from the left or right.

The familiar rules still hold: chain rule, product rule, and quotient rule all carry over. Complex differentiability implies continuity, but continuity does not imply differentiability.

Holomorphic functions

A function is holomorphic on a region if it's complex-differentiable at every point in that region. "Holomorphic" and "analytic" turn out to be equivalent in complex analysis, which is a deep and non-obvious result.

Key properties of holomorphic functions:

• They can be represented as convergent power series
• Their real and imaginary parts are harmonic functions (they satisfy Laplace's equation)
• They are infinitely differentiable (differentiable once means differentiable forever)

Standard examples: polynomials, $e^z$, $\sin(z)$, $\cos(z)$.

Cauchy's integral theorem

This is one of the central results in complex analysis:

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

for any closed contour $C$ in a simply connected domain where $f$ is holomorphic. "Simply connected" means the domain has no holes.

Think of it as a complex version of the fundamental theorem of calculus. The theorem says that for holomorphic functions, the integral around any closed loop is zero. This leads directly to path independence of integrals and opens the door to Cauchy's integral formula and the residue theorem.

Integration in the complex plane

Complex integration is where the subject really shows its power. Contour integrals can solve problems that are extremely difficult using real methods alone.

Contour integrals

A contour integral evaluates a complex function along a path (contour) in the complex plane. If the contour $C$ is parameterized by $z(t)$ for $t \in [a, b]$:

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

For analytic functions in simply connected domains, the integral depends only on the endpoints, not the specific path. This path independence is a direct consequence of Cauchy's integral theorem.

Residue theorem

The residue theorem is the workhorse of complex integration:

$\oint_C f(z)\,dz = 2\pi i \sum_{k=1}^n \text{Res}(f, a_k)$

It says that a contour integral around a closed curve equals $2\pi i$ times the sum of the residues at the poles enclosed by the contour. A residue at a pole $a_k$ is the coefficient of the $(z - a_k)^{-1}$ term in the Laurent series expansion around that pole.

To apply the residue theorem:

1. Identify all singularities of $f(z)$ inside the contour
2. Compute the residue at each singularity
3. Sum the residues and multiply by $2\pi i$

Applications to real integrals

One of the most striking uses of complex analysis is evaluating real integrals that resist standard techniques. The general strategy:

1. Extend the real integral into the complex plane by choosing an appropriate closed contour
2. Show that the contribution from the added portion of the contour vanishes (often using Jordan's lemma)
3. Apply the residue theorem to compute the closed contour integral
4. Extract the value of the original real integral

This works especially well for integrals of the form $\int_{-\infty}^{\infty} f(x)\,dx$, trigonometric integrals, and integrals with oscillatory behavior. Fourier and Laplace transforms also rely on these complex integration techniques.

Series and expansions

Series expansions let you represent complex functions in terms of powers of $(z - a)$. They're essential for understanding local behavior, classifying singularities, and computing residues.

Taylor series for complex functions

If $f$ is analytic at a point $a$, it has a Taylor series:

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

The series converges inside a disk centered at $a$. The radius of that disk extends to the nearest singularity. This is a stronger result than in real analysis, where a function can be infinitely differentiable yet not equal to its Taylor series.

Laurent series

When a function has a singularity at $a$, the Taylor series breaks down. The Laurent series handles this by allowing negative powers:

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

This converges in an annular (ring-shaped) region around $a$. The negative-power terms form the principal part, and they tell you what kind of singularity you're dealing with. The coefficient $a_{-1}$ is the residue at that point.

Singularities and poles

A singularity is a point where a function fails to be analytic. There are three types:

• Removable singularity: The function can be redefined at that point to become analytic. The Laurent series has no negative-power terms. Example: $\frac{\sin z}{z}$ at $z = 0$.
• Pole of order $n$: The principal part has finitely many terms, with the most negative power being $(z-a)^{-n}$. Example: $\frac{1}{z^2}$ has a pole of order 2 at $z = 0$.
• Essential singularity: The principal part has infinitely many terms. The function behaves wildly near the point. Example: $e^{1/z}$ at $z = 0$.

Classifying singularities is crucial for applying the residue theorem correctly.

Conformal mappings

A conformal mapping is an analytic function with a nonzero derivative. These mappings preserve angles between curves at every point, which makes them invaluable for transforming complicated geometric regions into simpler ones.

Preservation of angles

When two curves intersect in the complex plane, a conformal map preserves the angle between them at the point of intersection. This angle-preservation property comes directly from the fact that analytic functions with nonzero derivatives act locally as rotations and scalings.

Conformality breaks down at points where the derivative is zero or doesn't exist. At such points, angles can be distorted.

Types of conformal mappings

• Möbius transformations (linear fractional transformations): $w = \frac{az+b}{cz+d}$. These map circles and lines to circles and lines, and form the most fundamental class of conformal maps.
• Exponential and logarithmic maps: These transform between strips and sectors. For instance, $w = e^z$ maps horizontal strips to sectors.
• Power functions: $w = z^n$ maps a sector of angle $\alpha$ to a sector of angle $n\alpha$.
• Joukowski transformation: Used in aerodynamics to map circles to airfoil shapes.
• Schwarz-Christoffel transformations: Map the upper half-plane to polygonal regions.

Applications in physics

Conformal mappings are practical tools in physics and engineering because many physical problems (electrostatics, fluid flow, heat conduction) are governed by Laplace's equation, and solutions to Laplace's equation are preserved under conformal maps.

• Electrostatics: Transform complicated electrode geometries into simpler ones, solve the problem there, then map back
• Fluid dynamics: Simplify flow domains to find velocity fields and streamlines
• Heat conduction: Transform irregular boundaries into standard shapes where solutions are known
• Aerodynamics: The Joukowski transformation turns a circle into an airfoil, making lift calculations tractable

Complex analysis applications

The techniques of complex analysis appear throughout science and engineering, often in places you wouldn't expect.

Fluid dynamics

For irrotational, incompressible (two-dimensional) flow, the velocity field can be encoded in a single complex potential $w(z) = \phi + i\psi$, where $\phi$ is the velocity potential and $\psi$ is the stream function. Conformal mappings transform flow around simple shapes (circles) into flow around complicated ones (airfoils), which is exactly how the Joukowski transformation works.

Electrical engineering

• Complex impedance $Z = R + iX$ unifies resistance, capacitance, and inductance into a single framework for AC circuit analysis
• Transfer functions in control theory are functions of a complex variable $s$
• Fourier and Laplace transforms are built on complex integration and are fundamental to signal processing
• Complex power $S = P + iQ$ describes both real power dissipation and reactive energy storage

Quantum mechanics

Complex numbers aren't optional in quantum mechanics. Wave functions are inherently complex-valued, and the Schrödinger equation:

$i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi$

has $i$ built right into it. Hermitian operators have complex eigenfunctions, path integrals in quantum field theory use complex analysis techniques, and analytic continuation plays a role in defining scattering amplitudes.

Riemann surfaces

Riemann surfaces solve a fundamental problem: how do you handle functions like $\sqrt{z}$ or $\log(z)$ that give multiple outputs for a single input?

Multi-valued functions

Functions like $\sqrt{z}$, $\log(z)$, and $\arcsin(z)$ are multi-valued: a single input produces more than one output. For example, $\sqrt{1} = \pm 1$, and $\log(1) = 2\pi i n$ for any integer $n$.

This breaks the standard definition of a function. Riemann surfaces resolve the issue by creating separate "sheets," one for each branch of the function. The multi-valued function becomes single-valued when you let it live on this larger surface instead of the ordinary complex plane.

Branch points and cuts

A branch point is where the sheets of a Riemann surface connect. For $\sqrt{z}$, the branch point is at $z = 0$: if you walk around the origin in the complex plane, you end up on a different sheet.

A branch cut is a curve (often a ray from the branch point to infinity) that you draw to prevent accidentally crossing between sheets. The choice of branch cut is somewhat arbitrary, but once chosen, it determines which "branch" of the function you're working with. For the complex logarithm, the standard branch cut runs along the negative real axis.

Topology of Riemann surfaces

Riemann surfaces can have rich topological structure:

• The Riemann surface for $\sqrt{z}$ is topologically a sphere (genus 0)
• More complicated functions produce surfaces with "holes," classified by their genus (the number of holes)
• Compact Riemann surfaces are fully classified by their genus
• The uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three standard surfaces: the plane, the disk, or the sphere

Advanced topics

These topics push deeper into the theory and connect complex analysis to other areas of mathematics.

Analytic continuation

Analytic continuation extends an analytic function beyond its original domain of definition. The idea: if two analytic functions agree on an overlapping region, they must be the same function. So you can "grow" a function outward by finding overlapping power series expansions.

The classic example is the Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$, which converges only for $\text{Re}(s) > 1$ but can be analytically continued to the entire complex plane (except $s = 1$). The uniqueness of analytic continuation guarantees that this extension is the only consistent one.

Entire functions

An entire function is analytic on the whole complex plane with no singularities anywhere. Polynomials, $e^z$, $\sin(z)$, and $\cos(z)$ are all entire.

Two important results about entire functions:

• Liouville's theorem: If an entire function is bounded, it must be constant. This seemingly simple statement has a powerful consequence: it gives a clean proof of the fundamental theorem of algebra (every non-constant polynomial has a complex root).
• Weierstrass factorization theorem: Every entire function can be written as an infinite product, generalizing how polynomials factor into linear terms.

Meromorphic functions

A meromorphic function is analytic everywhere except at isolated poles. You can think of meromorphic functions as the "rational functions" of complex analysis: they can be expressed as ratios of entire functions.

• The poles of a meromorphic function are isolated (they don't cluster together in any finite region)
• On compact Riemann surfaces, meromorphic functions have particularly elegant properties
• Elliptic functions (doubly periodic meromorphic functions) and modular forms are important examples that connect complex analysis to number theory and algebraic geometry