Why can one ignore the real part of a complex number because of Fourier transform "only using the rotational part"?

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Why can one ignore the real part of a complex number because of Fourier transform "only using the rotational part"? Particularly, what property of Fourier transform is this?

Like given in this thread:

https://electronics.stackexchange.com/questions/78638/why-do-we-use-s-j-omega-in-ac-analysis-instead-of-s-sigmaj-omega


Essentially they say that (quoting clabacchio):

The reason why S=jω is chosen to evaluate AC signals is that it allows to convert the Laplace transform into Fourier transform.

The reason is that while S is a complex variable, what's used in the Fourier representation is just the rotational (imaginary) component, hence σ=0.