Using contour integration around a rectangle with vertices

$$-x, x, x+i y,-x+i y,$$

prove that, for all real $y$,

$$\int_{-\infty}^{+\infty} e^{-(x+i y)^{2}} d x=\int_{-\infty}^{+\infty} e^{-x^{2}} d x$$

Hence derive that the function $f(x)=e^{-x^{2} / 2}$ is an eigenfunction of the Fourier transform

$$\widehat{f}(y)=\int_{-\infty}^{+\infty} f(x) e^{-i x y} d x$$

i.e. $\widehat{f}$ is a constant multiple of $f$.