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.