There is nothing

*complex*about complex line integral. It is just vector addition (and taking a limit of that sum). Let’s take a concrete example: $$ \oint_{\lambda} \frac {1}{\zeta} d \zeta = 2 \pi i $$ Here, let \(\lambda \) be the unit circle centered at the origin. Then we pick \( \zeta \) from the circumference of the circle. Suppose we work with polar coordinates. Then the coordinate of a typical \( \zeta \) is \( (1, \theta) \).