# Inverse Fourier transform: $\int_{-\infty}^{+\infty}\exp\left(-{(x-x_0)^2\over2\sigma_x}\right)\exp\left(-i\phi\right)\exp(-ix t)dx$

To compute the inverse Fourier transform I need to evaluate the following integral $$\int_{-\infty}^{\infty}\exp\left(-{(\omega-\omega_0)^2\over2\sigma_\omega}\right)\cdot\exp\left(-i\phi\right)\cdot\exp(-i\omega t)\;\mathrm{d}\omega$$

Can I ask you guys for a hint how I should proceed? What are the common methods for tackling such problems?

You may just recall the gaussian integral result : $$\int_{-\infty}^{+\infty}\exp\left(-a{(x-x_0)^2}\right)\mathrm{d}x= \sqrt{\frac{\pi}{a}},\quad \Re a>0,\,x_0 \in \mathbb{C},$$ and rewrite your initial integral $$I=\int_{-\infty}^{+\infty}\exp\left(-{(\omega-\omega_0)^2\over2\sigma_\omega}\right)\cdot\exp\left(-i\phi\right)\cdot\exp(-i\omega t)\;\mathrm{d}\omega$$ as $$I=\exp\left(-i\phi\right)\cdot\exp(\frac{t^2\sigma_\omega}{2}-i\omega_0 t)\cdot\int_{-\infty}^{+\infty}\exp\left(-{(\omega+it\sigma_\omega-\omega_0)^2\over2\sigma_\omega}\right)\mathrm{d}\omega$$ leading to
$$I=\sqrt{2\pi\sigma_\omega}\cdot\exp\left(-i\phi\right)\cdot\exp(\frac{t^2\sigma_\omega}{2}-i\omega_0 t).$$