[Math] conversion of 2D Gaussian into polar coordinates

Question

Is it possible to convert the 2D Gaussian function in to polar coordinates?

$$\frac{1}{2\pi\sigma^2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp\big(-({(x-\mu_x)^2+(y-\mu_y)^2})/{2\sigma^2}\big) \,\mathrm{d}x\,\mathrm{d}y $$

Answer 1

Hint: There are many symmetries at work here. What if $\mu _x = \mu _y = 0$?