 Solution

Write $a=s+ib$ with real $s,b$, and first translate the real integration variable by $s$. It remains to integrate $e^{-z^2}$ along the horizontal line $\mathrm{Im}z=b$.

For $b>0$, apply the Cauchy integral theorem to the rectangle with vertices $-R,R,R+ib,-R+ib$. The integrand is an entire function, sois the negative of the two vertical-edge integrals. On either vertical edge, $z=\pm R+iy$ andTheir combined absolute value is at most $2b{e}^{-R^2+b^2}$ and tends to zero. The rectangle below the real axis gives the same estimate with $|b|$ when $b<0$; $b=0$ needs no deformation. Absolute convergence follows from $|e^{-(x+ib{)}^2}|=e^{b^2}{e}^{-x^2}$. Taking $R\to \infty$ proves the Gaussian contour translation identity