The stationary phase method deals with integrals such as $$I=\int_a^bf(t)e^{i\lambda g(t)}dt;$$ if there is a stationary point $c\in (a,b)$ then we can say that the above integral is approximately $$I\sim f(c)e^{i\lambda g(c)}\sqrt{\frac{2\pi}{\lambda|g''(c)|}}e^{\frac{\pi i}{4}}$$ as $\lambda\to\infty$. My question is as follows:
Can this method still be applied if $\lambda=1$ and if $g''(c)$ is nonzero but small?
I can't find a reference in the literature to answer my question, and naturally I expect the approximation to be less strong with the conditions I've mentioned, but is it still correct up to leading order?