polar coordinates of Gaussian Distribution with non zero mean

I found that the polar coordinates of 2-dimensional Gaussian distribution with mean zero $$\frac{1}{2\pi\sigma^2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp\big(-({x^2+y^2})/{2\sigma^2}\big) \,\mathrm{d}x\,\mathrm{d}y$$ is $$\frac{1}{\sigma^2}\int_{0}^{\infty}\exp\big(-r^2/{2\sigma^2}\big) \,r\mathrm{d}r$$ What if we consider non-zero mean, that is what would exactly be the following equation in polar coordinate system? $$\frac{1}{2\pi\sigma^2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp\big(-({(x-\mu_x)^2+(y-\mu_y)^2})/{2\sigma^2}\big) \,\mathrm{d}x\,\mathrm{d}y$$

If you mean a polar coordinate system with respect to the origin, then the result is a complicated mess. However, expressed in a polar coordinate system with respect to the point $(\mu_x,\mu_y)$, your third expression is again equal to your second expression, where $r$ now stands for the distance from the point $(\mu_x,\mu_y)$.
• Thanks a lot for your reply. Actually I am looking for overlapping probability of two 2D Gaussians centred at $(\mu_x,\mu_y)$. For the amount of overlap I need to consider the centre point of both the Gaussian functions, since both normally distributed points are located at some distance from each other – shaikh Jul 4 '11 at 11:36
• I don't know the term "overlapping probability" -- could you please define it? Also I'm not sure from what you wrote whether the two Gaussians are both centred at the same point $(\mu_x,\mu_y)$, or at different points. – joriki Jul 4 '11 at 12:00
• Assuming that there are two 2D Gaussians at $(\mu_{x1},\mu_{y1})$ and $(\mu_{x2},\mu_{y2})$ respectively. Since the Gaussian functions are infinite, I am interested in their overlapping area. – shaikh Jul 4 '11 at 12:04