I'm a Physics student (self-studying Zee's Quantum Field Theory in a Nutshell) hoping to get the following mathematical query solved.
It is mentioned that to approximate $I=\int_{-\infty}^\infty e^{-f(q)/\hbar} dq$, where $f$ is a real-valued function, the maximum contribution is from the minimum of $f$ for small $\hbar$. Assuming that $f$ has a (local as well as global) minimum at $q=a$, we expand $f$ around $q=a$ as $f(q) = f(a)+{1\over 2}f''(q-a)^2+O((q-a)^3)$ and then exploit the known Gaussian integral to get $$I\approx e^{-f(a)/\hbar}\Biggl({2\pi\hbar\over f''(a)}\Biggr)^{1/2}e^{-O(h^{1/2})}.$$
Question: What I don't understand is the origin of the term $e^{-O(h^{1/2})}$. Can you help?