Some questions concerning asymptotic expansions of integrals

[iccanobif]

I've started self-teaching asymptotic methods, and I have some theoretic questions (and lots of doubts!).

1. Say I have the asymptotic expansion

$f(x) \asymp \alpha \sum_n a_n x^{-n}$

for $x$ large, where $\alpha$ is some prefactor.

How can I estimate the value of $n$ for the term of least magnitude?

2. Suppose I have the integral

$I(\lambda) = \int_{a}^{b} f(t) \exp{(i\lambda \phi(t))} dt$,

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


Thanks!

 

[Redbelly98]

1. Two ways that come to mind are

• If a_n is defined explicitly, then treat n as if it were a continuous variable and use normal calculus methods to find the minima and maxima of a_nx^-n.
• Or, solve for n:
  a_nx^-n = a_n+1x^-(n+1)

2. You're correct that the integral would be small. Somebody else will have to chime in on how to figure out the order of magnitude.