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Is there a program that does Fourier transform, inverse fourier transform and fourier series?

Yes.

Help -> Tree -> (upward to “main” entry) Search “Fourier”

Results:
- fft
- ifft
- fourier_an, _bn, _cn (Fourier coefficients functions for cos, sin and e^i)

These are CAS functions.

Reference URL's

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pedrodebem

pinkman