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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 $$(1, \theta)$$.

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