Any possible interpretation Of Complex Integration When we studied Riemann Integration on the Real line we found that it gives us the area covered by that function. Also while studying line integral and Multiple integral we get a physical interpretation. What can one say about Complex Integration, I mean to say what are we essentially doing? Thanks in Advance....
 A: Complex integrals tend to be the integral of a function around some closed path $\gamma$:
$$\oint_\gamma f$$
Here, we can introdue a specific parametrization of $\gamma$ - in this example consider a circle, so $\gamma:[0,1]\to\mathbb C$ is defined by $\gamma(t) = Re^{2\pi it}$.
So, we have that:
$$\oint_\gamma = f(\gamma(t))\gamma'(t)dt = \int_0^{1} f(Re^{2\pi it}) (2\pi i R)e^{2\pi it}dt$$
Here, you might notice that this is really just a line integral, just expressed in complex coordinates.  That's all complex integration really is - line integrals, that are quite often around closed paths.
A: Under appropriate conditions, the Fundamental Theorem of Calculus applies so, with $F'=f$
and $\Gamma $ a path from $\alpha $ to $\beta $
$$
\int_\Gamma f(z)\,dz=F(\beta)-F(\alpha)
$$
In general the right-hand-side won't correspond to anything physically interpretable, it is just the difference in value of a complex function evaluated at two points.
If you adopt a parametrization of some path in the complex plane, $\gamma(t), t \in [a,b]$, then you can do things like compute the length of the path as
$$
\int_a^b |\gamma'(t)|\,dt
$$
but here you are doing 'special' things with the integrand.
