- #1

iccanobif

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1. Say I have the asymptotic expansion

[itex]f(x) \asymp \alpha \sum_n a_n x^{-n}[/itex]

for [itex]x[/itex] large, where [itex]\alpha[/itex] is some prefactor.

How can I estimate the value of [itex]n[/itex] for the term of least magnitude?

2. Suppose I have the integral

[itex]I(\lambda) = \int_{a}^{b} f(t) \exp{(i\lambda \phi(t))} dt [/itex],

for [itex]\lambda[/itex] large.

In the stationary phase method, if the function [itex]\phi(t)[/itex] has no stationary point in the interval [itex][a,b][/itex], am I wrong to believe that then [itex]I(\lambda)[/itex] is small beyond all orders (as the rapid oscillations of the phase imply cancellations)? How to formally derive the order of magnitude of [itex]I(\lambda)[/itex]?

Thanks!