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
.
References
[4]
[5]
[6]
- ↑ 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.
- ↑ Whaley, J., 2017, Oil in the Heart of South America, https://www.geoexpro.com/articles/2017/10/oil-in-the-heart-of-south-america], accessed November 15, 2021.
- ↑ Wiens, F., 1995, Phanerozoic Tectonics and Sedimentation of The Chaco Basin, Paraguay. Its Hydrocarbon Potential: Geoconsultores, 2-27, accessed November 15, 2021; https://www.researchgate.net/publication/281348744_Phanerozoic_tectonics_and_sedimentation_in_the_Chaco_Basin_of_Paraguay_with_comments_on_hydrocarbon_potential
- ↑ Alfredo, Carlos, and Clebsch Kuhn. “The Geological Evolution of the Paraguayan Chaco.” TTU DSpace Home. Texas Tech University, August 1, 1991. https://ttu-ir.tdl.org/handle/2346/9214?show=full.