I have no idea how to continue to find a solution for an Fourier transformation of the form $\exp\left\{-(x-a)^2\right\}$, this is what I tried (already found):
$$\mathcal{F}\{f\}(\omega)= \int_{-\infty}^{+\infty} \exp\left\{-(x-a)^2\right\}\exp\{-2\pi \textbf{i}\:\omega x\} \,dx$$ $$= \int_{-\infty}^{+\infty} \exp\left\{-(x-a)^2 -2\pi \textbf{i}\:\omega x\right\} \,dx $$ $$= \int_{-\infty}^{+\infty} \exp\{-x^2 +2ax -a^2 -2\pi \textbf{i}\:\omega x\} \,dx $$ $$= e^{-a^2}\int_{-\infty}^{+\infty} \exp\left\{-x^2 -x(2a-2\pi \textbf{i} \:\omega)\right\} \,dx $$
But here I have no idea what to do
With substitution $y=x-a$, you can "translate" it so that this helps:
What is the Fourier transform of $f(x)=e^{-x^2}$?