Isolated Singularities and Residue Theorem

First of all, I want to apologize for the names I'm going to use on this wiki, because many of them probably have different names when written in books. We'll suppose that $\Omega \subseteq \mathbb{C}$ is an open set throughout, unless otherwise stated, and $f \in \mathcal{H}\big(\Omega\big)$ means $f$ is a holomorphic function in $\Omega$. We are going to deal with integrals, series, Bernouilli numbers, Riemann zeta function, and many interesting problems, as well as many theories. It's very important to tell everything is very joined and connected inside of complex analysis, so we'll use much knowledge of complex analysis and all the branches of mathematics.

Relevant wiki: Contour Integration

Definition of Winding Number

If $\gamma : [0,1] \to \mathbb{C}$ is a closed piecewise smooth path and $a \not\in \gamma\big([0,1]\big)$, the winding number of $\gamma$ with respect to $a$ is $\text{ Ind}(\gamma , a) = \frac{1}{2\pi i} \cdot \int_{0}^{1} \frac{\gamma'(s)}{\gamma(s) - a} \, ds = \frac{1}{2\pi i} \oint_{\gamma} \frac{1}{z - a} \, dz.$ Intuitively, $\text{Ind}(\gamma, a)$ is the number of times $\gamma$ loops around $a$ anticlockwise.

Example 1

If $\gamma (x) = e^{2\pi i x}$ for $x \in [0, 1],$ its winding number about $0$ is $\text{ Ind}(\gamma , 0) = \frac{1}{2\pi i} \cdot \int_{0}^{1} \frac{2\pi i \cdot e^{2\pi i x}}{e^{2\pi i x}} \, dx = 1.$

Non-Example

If $\gamma_1 (x) = e^{\pi i x}$ for $x \in [0, 1],$

$\frac{1}{2\pi i} \cdot \int_{0}^{1} \frac{\pi i \cdot e^{\pi i x}}{e^{\pi i x}} \, dx = \frac{1}{2}.$

Nevertheless, we can't say $\text{Ind}(\gamma_1 , 0) = \frac{1}{2}$ because $\gamma_1$ is not closed.

The winding number of a path $\gamma$ with respect to $a$ is constant in the connected component of $\mathbb{C}\setminus \text{Image}(\gamma)$ containing (or not) $a$. This is, the winding number is 0 if $\gamma$ doesn't turn around $a,$ and it's constant if $\gamma$ turns around $a$. (Think about the winding number as the number of turns around...)

Example 2 (Generalization of Example 1)

If $C_r (t) = a + re^{it}$ for $t \in [0, 2\pi],$ then $\text{ Ind}(C_r, z) = \begin{cases} 1, & \text{ if } |z-a| < r \\ 0, & \text{ if } |z-a| > r \\ \text{undefined}, & \text{ if } |z-a|=r\end{cases}$