Argument (Angle): $\theta = \arg(z) = \tan^{-1}(\frac{y}{x})$. The principal argument is usually in $(-\pi, \pi]$.

Polar Form: $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$.

Complex Functions

A function $f(z)$ where $z$ is a complex variable. Can be written as $f(z) = u(x,y) + iv(x,y)$, where $u$ and $v$ are real-valued functions of two real variables $x$ and $y$.

Analytic Function

A complex function $f(z)$ is analytic (or holomorphic) at a point $z_0$ if it is differentiable not only at $z_0$ but also at every point in some neighborhood of $z_0$.

Entire Function: A function that is analytic everywhere in the complex plane (e.g., $e^z, \sin z, z^n$).
Singular Point: A point where $f(z)$ is not analytic.

Cauchy-Riemann Equations

A necessary condition for $f(z) = u(x,y) + iv(x,y)$ to be analytic is that its partial derivatives satisfy:

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

If these conditions hold and the partial derivatives are continuous, then $f(z)$ is analytic.

Milne-Thompson Method

Used to find an analytic function $f(z)=u+iv$ if either $u(x,y)$ or $v(x,y)$ is given. If $u(x,y)$ is given: 1. Find $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$. 2. Let $\phi_1(x,y) = \frac{\partial u}{\partial x}$ and $\phi_2(x,y) = \frac{\partial u}{\partial y}$. 3. Substitute $x=z, y=0$ into $\phi_1$ and $\phi_2$ to get $\phi_1(z,0)$ and $\phi_2(z,0)$. 4. $f(z) = \int [\phi_1(z,0) - i\phi_2(z,0)] dz + C$.

If $v(x,y)$ is given: 1. Find $\frac{\partial v}{\partial x}$ and $\frac{\partial v}{\partial y}$. 2. Let $\psi_1(x,y) = \frac{\partial v}{\partial x}$ and $\psi_2(x,y) = \frac{\partial v}{\partial y}$. 3. Substitute $x=z, y=0$ into $\psi_1$ and $\psi_2$ to get $\psi_1(z,0)$ and $\psi_2(z,0)$. 4. $f(z) = \int [\psi_2(z,0) + i\psi_1(z,0)] dz + C$.

Power Series

An infinite series of the form $\sum_{n=0}^\infty a_n (z-z_0)^n$.

Taylor Series: If $f(z)$ is analytic at $z_0$, then $f(z) = \sum_{n=0}^\infty \frac{f^{(n)}(z_0)}{n!} (z-z_0)^n$.

Laurent Series

An extension of the Taylor series for functions that are analytic in an annulus (ring-shaped region) but may have singularities inside the inner circle.

$$ f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n = \sum_{n=0}^\infty a_n (z-z_0)^n + \sum_{n=1}^\infty \frac{b_n}{(z-z_0)^n} $$

The coefficients are given by $a_n = \frac{1}{2\pi i} \oint_C \frac{f(\zeta)}{(\zeta-z_0)^{n+1}} d\zeta$, where $C$ is a contour in the annulus.

The second sum is called the principal part of the Laurent series.

Types of Isolated Singularities

If $z_0$ is an isolated singular point of $f(z)$:

Removable Singularity: The principal part of the Laurent series about $z_0$ contains no terms (e.g., $\frac{\sin z}{z}$ at $z=0$).
Pole of Order $m$: The principal part contains a finite number of terms, with the highest power being $\frac{b_m}{(z-z_0)^m}$ and $b_m \neq 0$. If $m=1$, it's a simple pole. (e.g., $\frac{1}{z^2}$ at $z=0$ is a pole of order 2).
Essential Singularity: The principal part contains infinitely many terms (e.g., $e^{1/z}$ at $z=0$).

Residue of $f(z)$ at a Pole

The residue of $f(z)$ at an isolated singular point $z_0$, denoted $\text{Res}(f, z_0)$, is the coefficient $b_1$ of the term $\frac{1}{z-z_0}$ in the Laurent series expansion.

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

Cauchy's Residue Theorem

If $f(z)$ is analytic inside and on a simple closed contour $C$, except for a finite number of isolated singularities $z_1,