I have got a two-dimensional gaussian distribution, where $\sigma_x = \sigma_y$ and $\mu_x = \mu_y = 0$.
$$ f(x,y) = \frac{1}{2\pi \sigma^2} e ^{-\left( \frac{x^2 + y^2}{2\sigma^2} \right)}$$
I would like to know the integral over this function. I'm basically looking for the $2$D analog of the error function. The area I want to integrate over is an ellipse centered around zeros.
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1$$
How do I get an analytical solution to this problem?