According to the encyclopedia of math, stationary phase predicts that the contribution of end points of the integral
$I(x) =\int_a^b f(x) \exp(i k \ p(x)) \ dx$
are of $O(k^{-1})$, where $k$ is a large parameter. The contribution of the end points are given as
$I_a(k)=\frac{i}{k \ p'(s)} \exp(i k \ p(a)) f(a)$
Is there a simple reference that discusses this result? The discussions of stationary phase I have found so far either assume infinite bounds, or only include results from the stationary points.
Copson's Asymptotic Expansions considers all three cases (no stationary points, a stationary point on the boundary, a stationary point inside the interval), with the assumption that the functions are complex analytic. For the first case, the idea is just to apply integration by parts to $$\frac 1 {i k} \int_a^b \frac {f(x)} {p'(x)} d(e^{i k p(x)}).$$ In terms of the steepest descent method, if $p'$ is positive on $[a, b]$, we can take the contour $[a, a + i \epsilon, b + i \epsilon, b]$, since $p(x)$ near $x_0$ is approximately $p(x_0) + p'(x_0) (x - x_0)$.