Overview
A contour integral is an integral of a function of a complex variable over a path in the complex plane.
{% \int_C f(z)dz %}
where {% C %} represents the curve over which the integration takes place.
Calculation
The typical calculation of a contour integral depends on the definite integral. If the path in the complex can be written as a function {% z(t) %} of a real variable, {% t \in \mathbb{R} %}, then the contour integral can be calculated as
{% \int_C f(z)dz = \int_a^b f(z(t))z'(t)dt %}