Im trying to get my head around complex integration/complex line integrals. Real integration can be thought of as the area under a curve or the opposite of differentiation. Thinking of it geometrically as the area under a curve or the volume under a surface in 3 dimension is very intuitive.
So is there a geometric way of thinking about complex integration? Or should I just be viewing it as process that reverses differentiation? Or has integration other meanings in complex analysis?
Here is an example. Could someone explain this to me? Here's the definition of the integral along a curve gamma in $\mathbb{C}$, parameterized by $w(t):[a, b] \to \mathbb{C}$.
$$\int_\gamma f(z) \mathrm{d}z = \int_a^b f[w(t)] w'(t) \mathrm{d}t$$
So I have that:
$\gamma$ is the unit circle with anti-clockwise orientation parameterized by $w:[0, 2\pi] \to \mathbb{C}$.
$w(t) = e^{it} = \cos(t) + i \sin(t)$.
So if we use the definition of the integral,
$$\int_\gamma f(z) \mathrm{d}z = \int^b_a f[w(t)]w'(t) \mathrm{d}t$$
and work this out, it comes to
$$\int^{2\pi}_0 i \mathrm{d}t = 2{\pi}i$$
So what does this $2{\pi}i$ represent? Does it mean anything geometrically, like if a regular integral works out to be 10 that means the area under the curve between 2 points is 10...