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:

$\int _{\gamma }f(z)\,dz=\int _{a}^{b}f(\gamma (t))\gamma '(t)\,dt$

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.