Solving: $\int_{-\infty}^{+\infty}e^{-2i\pi xy-\pi x^2/r^2}dx$

Question

I would like to solve:$$\int_{-\infty}^{+\infty}e^{-2i\pi xy-\pi x^2/r^2}dx$$

Where $i$ is the imaginary unit, and $y,r$ are constants. I've used Wolfram to get $re^{-\pi r^2 y ^2}$. Any way of solving it step-by-step?

Answer 1

Call this function $f(y)$ so $f(0)=r$ from the usual Gaussian integral. We need only show $f'(y)=-2\pi r^2 y f(y)$. Integrating by parts, $$f'(y)=-2\pi i\int_\mathbb{R}x\exp(-2\pi ixy-\frac{\pi x^2}{r^2})dx=-2\pi r^2y\int_\mathbb{R}\exp(-2\pi ixy-\frac{\pi x^2}{r^2})dx.$$