When faced with the task of controlling an integral containing a phase such as , consider using a change of variables (or a Taylor-type approximation) to replace with a simpler phase, such as a linear phase or a quadratic phase in normal form. This potentially allows one to use the tools of Fourier analysis or contour integration.
Suppose one wants to show that a one-dimensional integral is rapidly decreasing in , where is a bump function and is a smooth phase which has no stationary points on the support of (and in particular, is strictly monotone on this support). Then one can perform a change of variables to replace the integral with a Fourier integral , where is another bump function (which can be written explicitly in terms of , the change of variables function , and the Jacobian factor ). Since the Fourier transform of a bump function is rapidly decreasing, the claim follows.
Now suppose one wants to understand the one-dimensional integral , where now has one stationary point on the support of , say at the origin . Suppose also that is non-degenerate with . By Taylor expansion, near the origin. If we eliminate the contribution away from the origin (which is rapidly decreasing in by the previous example), and then perform a smooth change of variables , then one is faced with an integral of the form , where is a bump function with . To proceed further, we create an epsilon of room and use the explicit integral
(where we take a standard branch cut of the logarithm) which can be established by contour integration (or using the square and rearrange trick). One can then use "adding and subtracting" to write . The contribution of the main term is
In fact a full asymptotic expansion in powers of can be obtaind by refining this method, leading to the method of stationary phase.
Note also that the asymptotics are consistent with base times height heuristics: the (signed) amplitude of the integrand at the stationary point is , while the width of the interval where the phase is stationary (in the sense that only differs from by is about , as can be seen from Taylor expansion.
(Talk about how wave packet decompositions are used to analyse FIOs, or to analyse the restriction problem)
In higher dimensions, placing a stationary phase in normal form may require Morse theory.