I've started self-teaching asymptotic methods, and I have some theoretic questions (and lots of doubts!).(adsbygoogle = window.adsbygoogle || []).push({});

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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Some questions concerning asymptotic expansions of integrals

**Physics Forums | Science Articles, Homework Help, Discussion**