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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?

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    $\begingroup$ A power spectrum is the absolute value squared of a Fourier transform. $\endgroup$
    – anoniem
    Jul 26 at 11:01
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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.

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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 $$ e^{inx} = \cos (nx) + i \sin (nx),\qquad \cos (nx) = \frac{e^{inx}+e^{-inx}}{2},\qquad \sin (nx) = \frac{e^{inx}-e^{-inx}}{2i} $$

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