I think this is still a topic not explained clearly and mathematicly enough in many sources (if mentioned at all).
Currently I'm working through Fourier again to fresh this up a little bit for an upcoming quantum mechanics lecture. There is one basic thing I still don't really understand.
When I transform a function $$ f(t) $$ into the frequency domain, name it:
$$\tilde f(\omega) = Re[\tilde f(\omega)] + Im[\tilde f(\omega)]$$
our new function generally consists of a real and an imaginary part. The real part gives us information about the frequencies and their magnitude.
So all people say the imaginary part determines the phase shift of the corresponding frequencies, but they never go into detail. Why does the imaginary part represent the phase shift of my function $f(t)$? It obviously does, I tried it out myself. One can think of it in the way of Euler's formula (complex function in the polar form), and the imaginary part vanishes when there is no odd contribution, i.e. the sine becomes zero for a phase shift of multiples of $ \pi$.
Is there any mathematicly clean way to show why and in which way this is the case?
And I'm also really interested in how I can interpret this imaginary part (quantitative and qualitative).
Like imagine you see the graph of $Im[\tilde f (\omega)]$, what can you say about the phase shift of different frequencies or about the original $f(t)$?