Why do we ignore the real part of the transfer function while calculating Frequency response?

Question

The transfer function obtained from the differential equation is a function of $s $ which $x + iy$, then why do we ignore the real part while finding out the frequency response magnitude and phase. Why is frequency response $H$ considered as $H(i\omega) $ instead as $ H(\sigma + i\omega)$?

Answer 1

The frequency response is the output of the system when its input is a sinusoid of the given frequency, which is the exponential of an imaginary quantity.

The rest are calculations: Laplace-transform the sinusoid, compute the output in the frequency domain, and verify that its phase and magnitude at a frequency $\omega$ can indeed be found by looking at $H(i \omega)$.

Answer 2

This might not be quite right but isn't it because when we set $x = 0$ i.e. $ s = iy$ then our Laplace transform is simply a fourier transform, and it is a fourier transform that takes us into the frequency domain and hence will give us the frequency response.