Contour in Complex Analysis¶

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

Ex: $\begin{aligned} γ : [0, 1] & \to C t & ⇝ t + i t \end{aligned}$

Ex: $\begin{aligned} γ : [0, 2] & \to C t & ⇝ t + i t^{2} \end{aligned}$

Ex: $\begin{aligned} γ : [0, π] & \to C t & ⇝ e^{i t} \end{aligned}$

1. Types of Paths¶

Simple Path: A path is called is called simple if if it doesn't intersect except at the end points. $i . e .,$ $\forall t_{1}, t_{2} \in (a, b)$ $ $γ (t_{1}) \neq γ (t_{2}) \Leftrightarrow t_{1} \neq t_{2}$ $

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

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

Credits: Pinku Kumar, Pranav Kumar

Last modified on: 2023-01-04 23:20:05