Integral of a complex-valued function along a curve

Definition

Suppose $\gamma :[a,b]\to \mathbb{C}$ is a smooth curve whose image lies inside an open subset $U\subset \mathbb{C}$. Suppose $f:U\to \mathbb{C}$ is a continuous map. The integral of $f$ along $\gamma$ is defined as follows:

${\int }_{\gamma }f(z)dz=\underset{a}{\overset{b}{\int }}f(\gamma (t)){\gamma }^{\prime }(t)dt$

Using the chain rule, it is clear that the value of the integral does not depend on the choice $\gamma$ of parametrization; rather, it depends only on the orientation.