Complex $\Phi$ to prove the group velocity of a wavepacket I am reading a lecture note (author: Pr. Barton Zwiebach from the physics department of M.I.T., Course: 8.04 'Quantum Physics I', title:    de Broglie Wavelength and Galilean Transformations, Phase and Group Velocities, Choosing the Wavefunction for a Free Particle) and they say that in the expression $$\Psi = \int dk \Phi(k) e^{i(kx-\omega (k) t)}$$ taking $\Phi$ real or complex doesn't change the group velocity of the wavepacket.
They prove that the wavepacket moves at the velocity $$v_g = \frac{d \omega}{dk}_{|k_0}$$ for $\Phi$ real, using the stationary phase principle (and, in a second step, using a Taylor expansion of $\omega (k)$ around $k_0$), but not for $\Phi$ complex.
So I tried to do it on my own:
If $\Phi$ is complex then $\Phi (k) = |\Phi(k)|e^{iArg(\Phi(k))}$
Using the stationary phase principle, I get:
$$x-\frac{d \omega}{dk}_{|k_0}t +\frac{d (Arg(\Phi(k))}{dk}_{|k_0} =0$$
From there I don't know what to do...
Same for the Taylor expansion method, I add a complex phase to $e^{ikx}$ but then I don't know how to cancel that phase out to find the same result as for the real case.
 A: Suppose that we have waves with dispersion equation $\omega
=\omega(k)$. A right-going wave-packet of finite extent,
and with initial profile $\varphi(x)$, can be
Fourier analyzed to give
$$
\varphi(x)= \int_{-\infty}^{\infty} \frac {dk}{2\pi} \Phi(k)
e^{ikx}.
$$
At later times this will evolve to
$$
\varphi(x,t)= \int_{-\infty}^{\infty} \frac {dk}{2\pi} \Phi(k) e^{ikx-i\omega(k) t}.
$$
Let us suppose that $\Phi(k)$ is non-zero only for a narrow band of
wavenumbers around $k_0$, and that, restricted to  this narrow band,  we can approximate the full
$\omega(k)$ dispersion equation  by
$$
\omega(k) \approx \omega_0+ U(k-k_0).
$$
Thus
$$
\varphi(x,t)= \int_{-\infty}^{\infty} \frac {dk}{2\pi} \Phi(k)
e^{ik(x-Ut)-i(\omega_0-Uk_0)t}.
$$
Comparing this with the Fourier expression for the initial
profile, we find that
$$
\varphi(x,t)= e^{-i (\omega_0-Uk_0) t} \varphi(x-Ut).
$$
The pulse envelope  therefore travels at speed $U$. The individual wave crests,
on the other hand, move at the phase velocity
$\omega(k)/k$. The phase velocity  is encoded in the  $e^{-i (\omega_0-Uk_0) t}$ overall phase factor.
