Integral over exp(complex polynomial)

Question

hellooo, i want to calculate the integral $\int^\infty_{-\infty}e^{-a(x-x_0)^2+ikx}dx$ ($a>0$, $x_0$ and $k \in \mathbb{R}$) . Through substituting $x-x_0=y$ i've came to $e^{ikx_0}\int^\infty_{-\infty}e^{-ay^2+iky}dy$ but i dont know what to do now. I've tried applying Cauchy's integral theorem with the path being a semicircle from $0$ to $\pi$ with radius $R\rightarrow\infty$ around the origin of the complex plane. And since the associated complex function is holomorph on the whole complex plane the residue theorem doesnt work here. Im assuming that the solution is simple and i just dont see it. I hope somebody else can see it :^)