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

1. 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