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 C, parameterized by $w:[a, b] ->C$ \begin{equation} \int_\gamma f(z)dz = \int_a^b f(w(t)).w{(t)}'dt \end{equation}
So I have -
$\gamma$ is the unit circle with anti-clockwise orientation parameterized by $w:[0, 2\pi]\to C$
$w(t) = e ^{it} = Cos(t) + iSin(t)$
So if use the definition of the integral,
$\int_\gamma f(z)dz = \int^b_a f(w(t)).w{(t)}'dt$, and work this out it comes to
$\int^{2\pi}_0 i dt = 2{\pi}i$
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...