How to think about complex integration

[bobby2k]

Let's say you integrate a complex function along a curve. How do you visualize it? This is explaned very well in multivariate calculus in terms of work, or for instance the weight of the line of we integrate over the density etc..

But when we look at complex function I get this: The function F is a complex number, and then we multiply it with dz, this means that what we are doing informally is adding the product of complex numbers. But if we multiply complex number we take one of them, and rotate it with the angle of the other one, and scale its absolute value(its new absolute value is the product of both absolute values). Now I can not make sense of this. Can you? How do you informally look at a complex line integral?

 

[homeomorphic]

There are two ways to do that, both explained in Visual Complex Analysis.

You can get a free sample here, which includes the vector-field approach to understanding complex integrals in chapter 11.

http://usf.usfca.edu/vca//PDF/vca-toc.pdf

In his earlier chapter on integration he does picture exactly what you are talking about. Part of this is that he drew lots of pictures, so that you can actually see it on a page, rather than just trying to imagine it. You can imagine it with practice. But another part of it is that you need sort of a strategy to be able to visualize it, which he describes. In the end, you can explain a lot of complex integrals geometrically, this way. It is still somewhat strenuous to picture it, but quite possible with practice.