I’ve already had Fourier Series, now name it 0 as a prelude to a full tirade on Fourier Series.

The intuition is that like Taylor series for any analytic functions(continuously differentiable), we can transform(change coordinate) the function into Fourier Series to express, knowing that there is pros and cons. For periodic functions Taylor series is ill-suited, while Fourier Series is ideal. For example:

After steps of tedious deduction, we get

Enhance the understanding on Nov 29, 2022: