On stationary phase integrals

Let f(x) and g(x) be real functions defined on the interval [a, b], with f(x) at least twice continuously differentiable, f′(x) monotone increasing, and f(x) of bounded variation. We consider the exponential integral where e(t) denotes exp 2πit. The purpose of this note is to prove sharp forms of the well-known estimates: A: If f′(x) is nonzero on [a, b], then I has order of magnitude The constant of proportionality depends on the function g(x). B: If f′(x) changes sign at x = c with a < c < b, then