Asymptotic expansions

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. An oft seen example is linearization of more complicated expressions, but the methods are not confined to linearization. Some formulas such as in the Method of stationary phase for Fourier-like integrals, or the Method of steepest descent for Laplace-like integrals are examples of this.

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 ${\displaystyle O}$ and the small oh ${\displaystyle o}$ 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 ${\displaystyle O}$ and the small oh ${\displaystyle o}$ symbols.

Large ${\displaystyle O}$

We say that the function ${\displaystyle f(x)}$ is Large O of ${\displaystyle g(x)}$ as ${\displaystyle x\rightarrow x_{0}}$

${\displaystyle f(x)=O(g(x))\qquad }$ as ${\displaystyle \qquad x\rightarrow x_{0}\qquad }$ if ${\displaystyle \qquad |f(x)|=k|g(x)|\qquad }$ ${\displaystyle \qquad x\rightarrow x_{0}}$.

Small ${\displaystyle o}$

We say that ${\displaystyle f(x)}$ is small oh of ${\displaystyle g(x)}$ as ${\displaystyle x\rightarrow x_{0}}$

${\displaystyle f(x)=o(g(x))\qquad }$ as ${\displaystyle \qquad x\rightarrow x_{0}\qquad }$ if ${\displaystyle \qquad \lim _{x\rightarrow x_{0}}\left|{\frac {f(x)}{g(x)}}\right|=0}$.

A large parameter ${\displaystyle \lambda }$ or a small parameter ${\displaystyle \epsilon }$

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 ${\displaystyle \lambda \rightarrow \infty }$ or as a small parameter ${\displaystyle \epsilon \rightarrow 0}$.

Examples

We consider the following examples of a large parameter

Polynomial

${\displaystyle f(\lambda )=a_{n}\lambda ^{n}+a_{n-1}\lambda ^{n-1}+...+a_{0}+a_{-1}\lambda ^{-1}+a_{-2}\lambda ^{-2}+...+a_{-m}\lambda ^{-m}=O(\lambda ^{n})\qquad }$ as ${\displaystyle \qquad \lambda \rightarrow \infty \qquad }$ for all ${\displaystyle m,n>0}$.

If we consider the same polynomial, but written with epsilons instead of lambdas, because we are going to consider how the polynomial behaves as ${\displaystyle \epsilon \rightarrow 0}$

${\displaystyle f(\epsilon )=a_{n}\epsilon ^{n}+a_{n-1}\epsilon ^{n-1}+...+a_{0}+a_{-1}\epsilon ^{-1}+a_{-2}\epsilon ^{-2}+...+a_{-m}\epsilon ^{-m}=O(\epsilon ^{-m})\qquad }$ as ${\displaystyle \qquad \epsilon \rightarrow 0\qquad }$ for all ${\displaystyle m,n>0}$.

Exponential function

We consider the large parameter ${\displaystyle \lambda \rightarrow \infty }$

${\displaystyle e^{-\lambda }=o(\lambda ^{-n})\qquad }$ because exponential decays (grows) faster than any negative (positive) power of ${\displaystyle \lambda }$.

We may consider the small parameter ${\displaystyle \epsilon \rightarrow 0}$ or as the large parameter ${\displaystyle \lambda \rightarrow \infty }$

${\displaystyle e^{-\epsilon }=O(1)\qquad }$ as ${\displaystyle \qquad \epsilon \rightarrow 0\qquad }$ or ${\displaystyle \qquad e^{\frac {1}{\lambda }}=O(1)\qquad }$ as ${\displaystyle \qquad \lambda \rightarrow \infty }$. The last equality follows because ${\displaystyle \lambda ^{-1}=O(1)}$.

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 ${\displaystyle \sum _{k=0}^{\infty }u_{k}(x)}$ converges to the function ${\displaystyle f(x)}$ if the ${\displaystyle n-}$th partial sum ${\displaystyle \sum _{k=0}^{n}u_{k}(x)}$ obeys for all ${\displaystyle n>N}$ where ${\displaystyle N(\epsilon )}$ is a finite number and for all ${\displaystyle \epsilon >0}$

${\displaystyle \left|f(x)-\sum _{k=0}^{n}u_{k}(x)\right|<\epsilon }$.

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 ${\displaystyle \left\{\psi _{n}(\lambda )\right\}}$ such that

${\displaystyle \left\{\psi _{n+1}(\lambda )\right\}=o(\left\{\psi _{n}(\lambda )\right\})\qquad }$ as ${\displaystyle \qquad \lambda \rightarrow \infty }$ .

We say that ${\displaystyle f(\lambda )}$ has an asymptotic expansion with respect to the sequence of functions ${\displaystyle \left\{\psi _{n}(\lambda )\right\}}$ to the order ${\displaystyle N}$ if

${\displaystyle f(\lambda )-\sum _{n=0}^{N-1}a_{n}\psi _{n}(\lambda )=O(\psi _{N}(\lambda ))\qquad }$ as ${\displaystyle \qquad \lambda \rightarrow \infty }$

or if

${\displaystyle f(\lambda )-\sum _{n=0}^{N-1}a_{n}\psi _{n}(\lambda )=o(\psi _{N-1}(\lambda ))\qquad }$ as ${\displaystyle \qquad \lambda \rightarrow \infty }$.

If this is the case, we write

${\displaystyle f(\lambda )\sim \sum _{n=0}^{N-1}a_{n}\psi _{n}(\lambda )\qquad }$ as ${\displaystyle \qquad \lambda \rightarrow \infty }$.

If the condition holds for all ${\displaystyle N}$ then we write

${\displaystyle f(\lambda )\sim \sum _{n=0}^{\infty }a_{n}\psi _{n}(\lambda )\qquad }$ as ${\displaystyle \qquad \lambda \rightarrow \infty }$.

The reader should note that, in practice, asymptotic expansions do not converge. Now, if we could make the large parameter ${\displaystyle \lambda }$ 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

${\displaystyle \lambda ^{-(n+k)}=o(\lambda ^{-n})\qquad }$ as ${\displaystyle \lambda \rightarrow \infty \qquad }$ for all ${\displaystyle k>0}$, meaning that a function ${\displaystyle f(\lambda )}$ has an asymptotic expansion of order ${\displaystyle N}$ if

${\displaystyle f(\lambda )-\sum _{n=0}^{N-1}a_{n}\lambda ^{-n}(\lambda )=O(\lambda ^{-N})\qquad }$ as ${\displaystyle \qquad \lambda \rightarrow \infty }$

or if

${\displaystyle f(\lambda )-\sum _{n=0}^{N-1}a_{n}\lambda ^{-n}=o(\lambda ^{-(N-1)})\qquad }$ as ${\displaystyle \qquad \lambda \rightarrow \infty }$.

Alternatively, in the case of a small parameter ${\displaystyle \epsilon \rightarrow 0}$, these expressions take the form

${\displaystyle f(\lambda )-\sum _{n=0}^{N-1}a_{n}\epsilon ^{n}(\lambda )=O(\epsilon ^{N})\qquad }$ as ${\displaystyle \qquad \epsilon \rightarrow 0}$

or if

${\displaystyle f(\lambda )-\sum _{n=0}^{N-1}a_{n}\epsilon ^{n}=o(\epsilon ^{(N-1)})\qquad }$ as ${\displaystyle \qquad \epsilon \rightarrow \infty }$.

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

${\displaystyle I(\lambda )=\int _{a}^{b}{\frac {f(t)}{(i\lambda \phi ^{\prime }(t))}}(i\lambda \phi ^{\prime }(t))e^{i\lambda \phi (t)}\;dt=\left.{\frac {f(t)}{(i\lambda \phi ^{\prime }(t))}}e^{i\lambda \phi (t)}\right|_{a}^{b}-{\frac {1}{i\lambda }}\int _{a}^{b}{\frac {d}{dt}}\left[{\frac {f(t)}{\phi ^{\prime }(t)}}\right]e^{i\lambda \phi (t)}\;dt}$.

Applying this operation repetitively yields

${\displaystyle I(\lambda )=\left.\sum _{n=0}^{N-1}{\frac {(-1)^{n}e^{i\lambda \phi (t)}}{(i\lambda )^{n+1}}}\left[{\frac {1}{\phi ^{\prime }(t)}}{\frac {d}{dt}}\right]^{n}\left[{\frac {f(t)}{\phi ^{\prime }(t)}}\right]\right|_{a}^{b}+{\frac {(-1)^{N}}{(i\lambda )^{N}}}\int _{a}^{b}e^{i\lambda \phi (t)}{\frac {d}{dt}}\left[{\frac {1}{\phi ^{\prime }(t)}}{\frac {d}{dt}}\right]^{N-1}\left[{\frac {f(t)}{\phi ^{\prime }(t)}}\right]\;dt}$

as ${\displaystyle \lambda \rightarrow \infty }$.

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 ${\displaystyle O(\lambda ^{-1})}$ as ${\displaystyle \lambda \rightarrow \infty }$, whereas a stationary point is of an asymptotically more slowly decaying contribution of ${\displaystyle O(\lambda ^{-1/2})}$.

References

1. Bleistein, N. and Handelsman, R.A., 1986. Asymptotic expansions of integrals. Courier Corporation.
2. Bleistein, N. (1984). Mathematical methods for wave phenomena. Academic Press.
3. Erdélyi, A. (1956). Asymptotic expansions (No. 3). Courier Corporation.