# Difference between revisions of "Asymptotic expansions"

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<math> f(\lambda) - \sum_{n=0}^{N-1} a_n \epsilon^{n} = o(\epsilon^{(N-1)}) \qquad </math> as <math> \qquad \epsilon \rightarrow \infty </math>. | <math> f(\lambda) - \sum_{n=0}^{N-1} a_n \epsilon^{n} = o(\epsilon^{(N-1)}) \qquad </math> as <math> \qquad \epsilon \rightarrow \infty </math>. | ||

− | == | + | == Sources of asymptotic series == |

Asymptotic power series may be obtained as [[Dictionary:Taylor series | Taylor]] or [[Laurent's theorem |Laurent]] expansions, in | Asymptotic power series may be obtained as [[Dictionary:Taylor series | Taylor]] or [[Laurent's theorem |Laurent]] expansions, in | ||

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Another source of asymptotic series are obtained via the repetitive application of [[integration by parts]] to integrals, particular | Another source of asymptotic series are obtained via the repetitive application of [[integration by parts]] to integrals, particular | ||

to Fourier-like or Laplace-like integrals. | to Fourier-like or Laplace-like integrals. | ||

+ | |||

+ | === Repetitive integration by parts === | ||

+ | |||

+ | The first application of integration by parts (integrating | ||

+ | the exponential) yields | ||

+ | |||

+ | <math> I(\lambda) = \int_a^b \frac{f(t)}{(i \lambda \phi^\prime(t))} (i \lambda \phi^\prime(t)) e^{i \lambda \phi(t)} \; dt = \left. \frac{f(t)}{(i \lambda \phi^\prime(t))} e^{i \lambda \phi(t)} \right|_a^b - | ||

+ | \frac{1}{i\lambda} \int_a^b \frac{d}{dt} \left[\frac{f(t)}{\phi^\prime(t)} \right] e^{i \lambda \phi(t)} \; dt | ||

+ | </math>. | ||

+ | |||

+ | Applying this operation repetitively yields | ||

+ | |||

+ | <math> I(\lambda) = \left. \sum_{n=0}^{N-1} \frac{(-1)^n e^{i \lambda \phi(t)} }{(i\lambda)^{n+1} } | ||

+ | \left[\frac{1}{\phi^\prime(t)} \frac{d}{dt}\right]^n \left[\frac{f(t)}{\phi^\prime(t)} \right]\right|_a^b | ||

+ | + \frac{(-1)^N}{(i \lambda)^N } \int_a^b e^{i \lambda \phi(t)} \frac{d}{dt} \left[ \frac{1}{\phi^\prime(t)} \frac{d}{dt} \right]^{N-1} \left[\frac{f(t)}{\phi^\prime(t)} \right] \; dt </math> | ||

+ | |||

+ | as <math> \lambda \rightarrow \infty </math>. | ||

+ | |||

+ | This formal result assumes that all of the parts are sufficiently differentiable, and there are no divisions | ||

+ | by zero. The first term of the summation is <math> O(\lambda^{-1}) </math> as <math> \lambda \rightarrow \infty </math>, whereas a stationary point is of an asymptotically more slowly decaying contribution | ||

+ | of <math> O(\lambda^{-1/2}) </math>. | ||

== References == | == References == | ||

{{reflist}} | {{reflist}} |

## Latest revision as of 16:24, 27 December 2020

Asymptotic analysis^{[1]}^{[2]}^{[3]} is the study of a function in the neighborhood of a point. That point need not be in the domain of the function.

This topic is relevant to the physical sciences because asymptotic expansions, or rather approximations consisting of the leading order
term(s) of the asymptotic series appear in many places in mathematical physics. In particular, physicists may use the term *semi-classical*
analysis* to label such approximations. *

Tacitly, asymptotic approximations exist in mathematical derivations where higher order terms are assumed to be *small*. Usually, this is
called a *linearization*, but the success of these linearizations fell
Some formulas such as in the Method of stationary phase for Fourier-like integrals, or the Method of steepest descent for
Laplace-like integrals.

Asymptotic series, in general, do not converge, so more terms are not guaranteed to provide a better approximation to a function.

# Bachman-Landau *Large Oh* and the *small oh* order symbols

A common way of expressing the order, which is to say the behavior of a function as it approaches a point is through
the *Large Oh* and the *small oh* symbols.

## Large

We say that the function is *Large O* of as

as if .

## Small

We say that is *small oh* of as

as if .

## A large parameter or a small parameter

In the physical sciences such parameters as frequency, wavenumber, or distance may be considered *large parameters* or as *small*
parameters. This means that we consider behavior of a function as a large parameter or
as a small parameter .

## Examples

We consider the following examples of a large parameter

### Polynomial

as for all .

If we consider the same polynomial, but written with epsilons instead of lambdas, because we are going to consider how the polynomial behaves as

as for all .

### Exponential function

We consider the large parameter

because exponential decays (grows) faster than any negative (positive) power of .

We may consider the small parameter or as the large parameter

as or as . The last equality follows because .

## Asymptotic series

A student normally learns about *convergent series* in the first or second year of their undergraduate mathematics program. In that program
the student learns that the infinite series of functions converges to the function if the th partial sum obeys for all where is a finite number and for all

.

We can create something similar which we call an *asymptotic expansion*. We develop this in terms of a large parameter, but these results
extend to the case of a small parameter as well. We consider a sequence of functions such
that

as .

We say that has an asymptotic expansion with respect to the sequence of functions to the order if

as

or if

as .

If this is the case, we write

as .

If the condition holds for all then we write

as .

The reader should note that, in practice, asymptotic expansions do not converge. Now, if we could make the large parameter arbitrarily large, then we could always find a value large enough that the series would converge. However in physical problems we have constraints impose on the values of parameters making them finite.

### Asymptotic power series

The most common asymptotic expansions encountered in the physical sciences are *asymptotic power series*. For example we note
that

as for all , meaning that a function has an asymptotic expansion of order if

as

or if

as .

Alternatively, in the case of a small parameter , these expressions take the form

as

or if

as .

## Sources of asymptotic series

Asymptotic power series may be obtained as Taylor or Laurent expansions, in terms of a large or small parameter.

Another source of asymptotic series are obtained via the repetitive application of integration by parts to integrals, particular to Fourier-like or Laplace-like integrals.

### Repetitive integration by parts

The first application of integration by parts (integrating the exponential) yields

.

Applying this operation repetitively yields

as .

This formal result assumes that all of the parts are sufficiently differentiable, and there are no divisions by zero. The first term of the summation is as , whereas a stationary point is of an asymptotically more slowly decaying contribution of .