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The most important calculation in all of complex analysis is the following:

$$ \int_C \frac{1}{z}dz = 2\pi i $$

Where $C$ is the unit circle centered at the origin. Let's see if we can understand this integral geometrically. Hopefully this will illuminate the general meaning of the complex integral as well.

Cut the unit circle up into a million tiny directed line segments (all pointed in the "positive" direction). Each one of these is a small change in $z$ along the curve: each one of these is a little "$dz$". The directed line segment from the origin to that point on the unit circle is $z$. For points on the unit circle, $\frac{1}{z} = \overline{z}$ is the reflection of $z$ in the real axis. Remember that complex numbers multiply by multiplying the moduli and summing the arguments. Since if the argument of $z$ is $\theta$, then the argument of $dz$ is $\frac{\pi}{2}+\theta$ by geometry (radius is perpendicular to tangent of a circle), and the argument of $\frac{1}{z}$ is $-\theta$. So the argument of $\frac{1}{z}dz$ is $\frac{\pi}{2}$! Also since $|z|=1$, $|\frac{1}{z}dz|=|dz| = ds$, the length of the little line segment. Thus $\frac{1}{z}dz$ is a little line segment of length $ds$ pointing straight up, where $ds$ is the length of a little arc of the circle. In other words, $\frac{1}{z}dz = i ds$ Summing all of these around the unit circle we get $2\pi i$.

Hopefully this picture helps you to visualize all of this: