I have a question concerning the convergence of the saddle point method. Applying the method to a function $\rho(x)$, defined as
\begin{align} \rho(x) = \frac{1}{2\pi}\int dk\, e^{\kappa f(k,\sigma)}, \end{align}
is it possible to determine a radius of convergence in terms of the function $f(k,\sigma)$ so that the asymptotic result
\begin{align} \rho(x) \underset{\kappa \to \infty}{\sim} e^{\kappa F(x,\sigma)} \end{align}
always holds? My goal is to set bounds to $\sigma$ indirectly, which is just a real parameter that appears in both functions $F(x,\sigma)$ and $f(k,\sigma)$, up to which $\sigma$ can be varied, without affecting the method. I took a look in large deviation theory, but nothing crossed my path that could shed any light on my problem.