# Some questions concerning asymptotic expansions of integrals

• iccanobif
In summary, the conversation discusses asymptotic methods and how to estimate the value of n for the term of least magnitude in an asymptotic expansion. It also touches on the stationary phase method and how to determine the order of magnitude for an integral with a rapidly oscillating phase.
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!

1. Two ways that come to mind are
• If an 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 anx-n.
• Or, solve for n:
anx-n = an+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.

## 1. What is an asymptotic expansion?

An asymptotic expansion is a mathematical tool used to approximate a function as a series of terms, with each term becoming increasingly smaller as the variable approaches a certain value. This allows for a more accurate estimation of the function's behavior near the variable of interest.

## 2. How are asymptotic expansions used in integrals?

Asymptotic expansions are used in integrals to simplify the calculation of the integral, especially for functions that cannot be integrated analytically. By using an asymptotic expansion, the integral can be approximated as a series of terms, which can then be integrated term by term.

## 3. What is the difference between a regular and a singular asymptotic expansion?

A regular asymptotic expansion is used for functions that are smooth and have no singularities. It is also known as a Taylor series expansion. A singular asymptotic expansion, on the other hand, is used for functions that have singularities, such as poles or branch points. In this case, the expansion must be modified to account for the singular behavior of the function.

## 4. How can one determine the accuracy of an asymptotic expansion?

The accuracy of an asymptotic expansion can be determined by calculating the remainder term, which is the difference between the exact value of the function and the truncated series. The smaller the remainder term, the more accurate the approximation will be.

## 5. Can asymptotic expansions be used for all integrals?

No, asymptotic expansions are not suitable for all integrals. They are most effective for integrals that have a large parameter or variable, or for integrals that have a complicated integrand. In some cases, the use of asymptotic expansions may lead to incorrect results, so caution must be taken when applying this method.

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