I have to calculate a pretty ugly integral which is given by: $$ \psi(x,t)=\int_{-\infty}^{\infty}\exp\left(-\frac{(p-p_0)^2}{4\,\Delta p^2} - \frac{i}{\hbar}(p-p_0)x_0+\frac{i}{\hbar}(px-\frac{p^2}{2m}t)\right)\mathrm{d}p $$
It's a solution to the Schrodinger equation and $x_0,p_0$ and $\Delta p$ are constants.
I tried to complete the square on this one, but for me it doesn't seem that the integral gets easier by that. I would be really grateful if someone could give me a good starting point in solving this integral. Maybe completing the square is a good way to do it, if this the case, I will try to do it again with that method.
Hint:
The exponent is a quadratic polynomial with some complex coefficients, so the integrand has the form
$$e^{ap^2+2bp+c}$$ where $a$ is real and negative.
You can get rid of $a$ by a scaling of the variable, and of the real part of $b$ by shifting the variable. The constant coefficient can be factored away. You end-up with an integral for the form
$$\int_{-\infty}^\infty e^{-p^2}(\cos\omega p+i\sin\omega p)\,dp$$ which is a classical one. (The imaginary part vanishes.)
https://www.wolframalpha.com/input/?i=integrate+e%5E%28-p%5E2%29+cos%28omega+p%29+from+-+inifinity+to+infinity