Where are the imaginary numbers in Fourier analysis hiding?

Question

Someone (in an ancient closed thread) said, "Fourier analysis is simplified if complex numbers are used."

Hey, wait, you separate the frequency components of a wave to represent it as a power spectrum. Where does $\sqrt{-1}$ appear in all this? Is it deep in the arithmetic of the FFT software, or does $i$ have some observable influence on the transformed wave?

Answer 1

Euler's formula tells us that

$$e^{i\theta} = \cos \theta + i \sin \theta$$

Using complex numbers allows us to handle $\sin$ and $\cos$ waves at the same time in areas such as Fourier analysis. It also suggests conceptual links between, for example, Fourier transforms and Laplace transforms.

Answer 2

What Gandalf said.
Real Fourier series look like $$ \sum_{n=0}^\infty a_n \cos(nx) + \sum_{n=1}^\infty b_n\sin(nx) $$ whereas complex Fourier series look like $$ \sum_{n=-\infty}^\infty c_n e^{i n x} $$ We can convert from one to the other using $$ e^{inx} = \cos (nx) + i \sin (nx),\qquad \cos (nx) = \frac{e^{inx}+e^{-inx}}{2},\qquad \sin (nx) = \frac{e^{inx}-e^{-inx}}{2i} $$