One thing I find fascinating about Fourier analysis is the way the trigonometric Fourier series played such a central role in “breaking mathematics”[1] and the crisis that led to providing a rigorous basis for limits and continuity and all the other stuff that is now called real and complex analysis.

Cauchy had just proved that the limit of the sum of an infinite set of continuous functions was itself continous, and then along came Fourier with “are you sure about that bro?” and showed that you could take the infinite sum of very clearly continuous functions (just sine and cosine) and approximate something like a sawtooth function (which was very obviously discontinous) as closely as you like.

[1] by which I mean making obvious the fact that they had been proceeding for 100+ years using calculus without a rigorous basis.