Integral along a Curve

Definition

We will define the integral along a curve $C$ in the complex plane. We will consider curves $C$ which admit a Smooth Parametrization.

Let $C$ be a smooth curve with parametrization $\gamma :[a,b]\to \mathbb{C}$. Let $f$ be a function which is continuous on $C$. Then we set
${\int }_{C}f(z)dz:={\int }_a^{b}f(\gamma (t)){\gamma }^{\prime }(t)dt.$

Example

An example can be found in Example of Integrating a Complex Curve.