Error Bounds for Stationary Phase Approximations

An error theory is constructed for the method of stationary phase for integrals of the \[I(x) = ∫_a^b {eⁱˣᵖ⁽ᵗ⁾ q(t)dt.} \]Here x is a large real parameter, the function $p(t)$ is real, and neither $p(t)$ nor $q(t)$ need be analytic in t. For both finite and infinite ranges of integration, explicit expressions are derived for the truncation errors associated with the asymptotic expansion of $I(x)$. The use of these explicit expressions for the computation of realistic error bounds is illustrated by means of an example.