Solution
= Solution
For a simple stationary point at the endpoint $b$, only one half of the local quadratic neighborhood lies in the integration interval. The even Gaussian integrand therefore gives half the interior <stationary phase> contribution:
$$
\boxed{I(\lambda)=f(b)\sqrt{\frac\pi{2\lambda|g''(b)|}}e^{i\lambda g(b)+i\pi\operatorname{sgn}(g''(b))/4}+O(\lambda^{-1}).}
$$