Contour in Complex Analysis

Def: A path is a continuous funtion $\gamma :[a,b] \subseteq \mathbb{R} \to \mathbb{ C }$. \(\gamma(t) = x(t) + i y(t)\)

Ex: \(\begin{align} \gamma: [0,1] &\to \mathbb{ C }\\ t &\leadsto t+it \end{align}\)

Ex: \(\begin{align} \gamma: [0,2] &\to \mathbb{ C }\\ t &\leadsto t+it^2 \end{align}\)

Ex: \(\begin{align} \gamma: [0,\pi] &\to \mathbb{ C }\\ t &\leadsto e^{it} \end{align}\)

Types of Paths

Simple Path: A path is called is called simple if if it doesn’t intersect except at the end points. $i.e.,$ $\forall t_1, t_2 \in (a,b)$ \(\gamma(t_1) \neq \gamma(t_2) \Leftrightarrow t_1 \neq t_2\)

Smooth Path: A path is called smooth if it is continuously differentiable and has non-zero derivatives, $i.e.,$ \(\gamma \in \mathcal{C}^1; \quad \gamma(t) \neq 0.\)

Closed Path: A path is closed if the end points are joined, $i.e.,$ \(\gamma(a) = \gamma(b).\)

Credits: Pinku Kumar, Pranav Kumar