Asymptotic analysis 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
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.
We say that the function is Large O of as
as if .
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 .
We consider the following examples of a large parameter
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 .
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 .
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
We say that has an asymptotic expansion with respect to the sequence of functions to the order if
If this is the case, we write
If the condition holds for all then we write
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
as for all ,
meaning that a function has an asymptotic expansion of order if
Alternatively, in the case of a small parameter , these expressions take the form
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
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
- ↑ Bleistein, N. and Handelsman, R.A., 1986. Asymptotic expansions of integrals. Courier Corporation.
- ↑ Bleistein, N. (1984). Mathematical methods for wave phenomena. Academic Press.
- ↑ Erdélyi, A. (1956). Asymptotic expansions (No. 3). Courier Corporation.