Math 185: Complex Analysis
Table of Contents
Lecture 1: Introduction (January 21, 2026)
Complex analysis is especially useful for computing some integrals that are difficult in $\mathbb R$, but easily computed in $\mathbb C$. This is useful for all sorts of domains, including mechanics, statistics (integrating the CDF of a normal distribution), and combinatorics.
Definition (Extended Complex Plane). Just the complex plane and infinity. Formally, $\hat{\mathbb{C}} = \mathbb{C} \cup \set{\infty}$.
Can be modeled by Riemann sphere.
Lecture 2: Operations on $\mathbb{C}$ (January 23, 2026)
$\mathbb{C} = \set{a + bi; a, b \in \mathbb{R}}$, where $i^2 = -1$. This is isomorphic to $\set{(a, b); a, b, \in \mathbb{R}}$, which is isomorphic to $\mathbb{R}^2$. In general, a complex number $x + yi$ can be thought of as a linear combination of two basis vectors, $1$ and $i$. (Equivalently, a complex number $(x, y)$ is the linear combination $x(1, 0) + y(0, 1)$.)
Addition on $\mathbb{C}$ is performed coordinate-wise.
Definition (Real and Imaginary Parts). For $z \in \mathbb{C}$, $z = x+yi$, $\text{Re}(z) = x$ and $\text{Im}(z) = y$.
A complex number can be purely real or purely imaginary.
Multiplication on $\mathbb{C}$
Definition (Multiplication of Complex Numbers). $(a_1, b_1) \times (a_2, b_2) = (a_1a_2 - b_1b_2, a_1b_2 + a_2b_1)$. This is equivalently (and more easily) defined by the following properties: $1 \times 1 = 1$, $1 \times i = i$, $i \times i = -1$.
We might care to define a multiplicative inverse of $z$ such that $z \times z^{-1} = 1$.
Definition (Multiplicative Inverse). If $z = x + yi$, then $\text{Re}(z^{-1}) = \frac{x}{x^2 + y^2}$ and $\text{Im}(z^{-1}) = \frac{-y}{x^2 + y^2}$. Notably, this is only defined when $z \neq 0$.
It follows that $\mathbb{C}$ is a field! It is also algebraically closed, which is essentially the Fundamental Theorem of Algebra.
Proposition. For any $z \in \mathbb{C}$, there exists $w \in \mathbb{C}$ such that $w^2 = z$.
Polar Coordinates
A complex number $z$ can be equivalently represented by $\vert {z} \vert = \sqrt{x^2 + y^2}$ and $\text{arg}(z) \in [0, 2 \pi)$, where $\text{arg}(z)$ is essentially the angle between the positive $\mathbb{R}$-axis and $z$. The argument is only defined for $z \neq 0$.
Lecture 3: Multiplication and Polar Coordinates (January 26, 2026)
What is the argument of $z + w$?
Polar coordinates are expecially useful for multiplication, $z \cdot w = r_1 \cdot r_2(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2))$. So $\vert z \cdot w \vert = r_1 \cdot r_2$. And $\text{arg}(z\cdot w) = \text{arg}(z) + \text{arg}(w) \pmod{2\pi}$.
This structure looks familiar! Notably, the logarithm behaves similarly: $\ln(ab) = \ln(a) + \ln(b)$. This will eventually motivate the complex exponential function: $e^{i \theta} = \cos \theta + i \sin \theta$.
What is multiplication?
Multiplication by 2, for example, scales the magnitude of the vector in the complex plane by 2.
Multiplying by $i$ is a rotation by $\frac{\pi}{2}$ anti-clockwise about the origin.
Lecture 4: Complex Functions (January 28, 2026)
In general, $\text{Re}(z) = \frac{z + \bar z}{2}$ and $\text{Im}(z) = \frac{z - \bar z}{2i}$.
The Complex Exponential Function
The exponential function, $f(x) = e^x$, is well-studied over real-valued inputs. It adheres to properties such as:
- $e^{x + y} = e^xe^y$.
- Strictly increasing and positive.
- $f’(x) = e^x$ (the function is its own derivative!).
How to define $e^z$, where $z$ might be a complex number with an imaginary part? Clearly, $e^z = e^{\text{Re}(z)}e^{i\text{Im}(z)}$, but what is $e^{i\text{Im}(z)}$?
Definition (Complex Exponential Function). $e^{iy} = \cos(y) + i \sin(y)$.
Complex Functions
A complex function $f$ acts on $\mathbb{C}$. If it is also defined over real-values, $\mathbb{R}$, the definition of $f$ for $\mathbb{C}$ must be consistent with the restriction of $f$ to $\mathbb{R}$. It is easy enough to verify that $e^z$ when $z$ is real is undisturbed by our definition for $e^z$ for complex-valued inputs.
Other complex functions include $\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}$ and $\cos(z) = \frac{e^{iz} + e^{-iz}}{2}$. It should be easy enough to verify that classic identities and properties of trigonometric functions still hold (for example, $\sin^2(z) + \cos^2(z) = 1$). (Notice that $\vert \cos(z)\vert$ may very well be $>1$! Consider $z = 3i$.)
Lecture 5: Complex Inverses (January 30, 2026)
Over real-valued inputs, $f(x) = e^x$ is a bijective map $\mathbb{R} \to {\mathbb{R}}^{>0}$. However, over complex-valued inputs, $e^0 = e^{2 \pi i}$, so $e^x$ is no longer injetive, and therefore no longer bijective. In order to still define an inverse function $f^{-1}(x)$, we restrict our consideration of $f(x)$ to $f(x)$ on some specific domain. Specifically, let $A_{y_0} = \set{z \in \mathbb{C}: y_0 \leq \text{Im}(z) < y_0 + 2\pi}$. Over inputs from $A_{y_0}$, $e^x$ is a bijective map to $\mathbb{C} \backslash \set{0}$.
Definition (Principle Branch of Logarithm). $\text{Log}(z): \mathbb{C} \backslash \set{0} \to A_0$. In other words, the inverse of $e^x: A_0 \to \mathbb{C} \backslash \set{0}$.