Let $u = u(x,y)$ be a smooth scalar field on $\mathbb R^2$ with a global minimum at the origin,
$u(0,0)$ =: $u_0<u(x,y)$. By Taylor expanding $u$ to second order around the origin, show that
one has the approximate identity
$\int_{R^2}$ exp$[-\lambda u(x,y)]d^2x$ $\approx \frac{2\pi}{\lambda}\frac{e^{-\lambda u_0}}{\sqrt {k_1 k_2}}$
The parameters $k_1$ and $k_2$ are the principal curvatures of $u$ at the origin, and $\lambda$ is an arbitrary
positive real number. Which parameter governs the quality of the approximation.
By convention, if $u(x,y)$ is just continuous, then we say $u(x,y) \in C^{(0)}(\Omega)$.
Also, $u(x,y) \in C^{(\infty)}$ if the function is differentiable any number of times. For instance, $e^{x} \in C^{(\infty)}$. I am confused the function itself where I have to Taylor expand to the second order. Any help or guide to this problem will be appreciated.