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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.

  1. 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?

  2. 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...

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    This seems to be a duplicate of http://math.stackexchange.com/questions/110334/line-integration-in-complex-analysis/110367#1103672012-02-20
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    I am asking for an explanation in 'plain english' (as much as possible) in this post as the answers in the other post were not intuitive to me. And specifically in this post I am asking what the result that I got - $2{\pi}i$ - actually means. If I got the derivative of a y = f(x) and it turned out to be 2 I know that that means y increase by 2 when x increases by 1. So what does the $2{\pi}i$ result that I got mean in the case of complex integration.2012-02-20
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    I suggest that you simply stop worrying about this. Reason by analogy with what you know, and try to transfer tentatively your intuition about the real integral to the complex one—with some aspects you should have absolutely no trouble, and with others you will. When you hit one of this problematic pieces of intuition, then experience, practice and knowledge will construct something in your mind that you will only be able to describe as intuition.$$ $$Trying to wrap one's head around something one is not familiar enough and about which one is only beginning to learn is a very unnatural thing!2012-02-21

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