>>10879094^ see that pic? Consider the integrals as being "dot products", we are simply expressing f(x) in terms of a sum of sines and cosines (in other words, our orthogonal basis)
It's actually amazing to see, just like Taylor series can approximate any function in terms of a sum of powers, Fourier series can express any function in terms of a sum of sines and cosines.
Now fourier series only work on a limited interval of the x-axis. So perhaps from -10 to 10, or -100 to 100. However, if we stretch that interval to infinity, we arrive at fourier transforms.
We decompose the sines and cosines into exponentials, and push the limits to infinity. By doing this, the sum turns into an integral, and we get pic related. These are Fourier transforms, they look like these two complementary integrals in pic related.
They are VERY important, especially in signal analysis and other such things.
Anyway, this is just some important background to have before you get into complex analysis.
The book i learned from was from Boas "Mathematical Methods on the Physical sciences".
You should get familiar with complex numbers, and complex functions. And then look into residue theorem, Cauchy Integral theorem, Laurant series.....etc.
