# Asymptotic analysis

﻿
Asymptotic analysis

In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are

The simplest example is, when considering a function f(n), there is a need to describe its properties when n becomes very large. Thus, if f(n) = n2+3n, the term 3n becomes insignificant compared to n2 when n is very large. The function "f(n) is said to be asymptotically equivalent to n2 as n → ∞", and this is written symbolically as f(n) ~ n2.

## Definition

Formally, given complex-valued functions f and g of a natural number variable n, one writes

$f \sim g \quad (\text{as } n\to\infty)$

to express the fact, stated terms of little-o notation, that

$f(n) = g(n) + o(g(n))\quad(\text{as }n\to\infty),\,$

or equivalently

$f(n) = (1+o(1))g(n)\quad(\text{as }n\to\infty).$

Explicitly this means that for every positive constant ε there exists a constant N such that

$|f(n)-g(n)|\leq\epsilon|g(n)|\qquad\text{for all }n\geq N~$.

Unless g(n) is infinitely often zero (which would make the limit below undefined), this statement is also equivalent to

$\lim_{n\to\infty} \frac{f(n)}{g(n)} = 1.$

This relation is an equivalence relation on the set of functions of n. The equivalence class of f informally consists of all functions g which are approximately equal to f in a relative sense, in the limit.

## Asymptotic expansion

An asymptotic expansion of a function f(x) is in practice an expression of that function in terms of a series, the partial sums of which do not necessarily converge, but such that taking any initial partial sum provides an asymptotic formula for f. The idea is that successive terms provide a more and more accurate description of the order of growth of f. An example is Stirling's approximation.

In symbols, it means we have

fg1

but also

fg1 + g2

and

$f \sim g_1 + \cdots + g_k$

for each fixed k, while some limit is taken, usually with the requirement that gk+1 = o(gk), which means the (gk) form an asymptotic scale. The requirement that the successive sums improve the approximation may then be expressed as $f - (g_1 + \cdots + g_k) = o(g_k).$

In case the asymptotic expansion does not converge, for any particular value of the argument there will be a particular partial sum which provides the best approximation and adding additional terms will decrease the accuracy. However, this optimal partial sum will usually have more terms as the argument approaches the limit value.

Asymptotic expansions typically arise in the approximation of certain integrals (Laplace's method, saddle-point method, method of steepest descent) or in the approximation of probability distributions (Edgeworth series). The famous Feynman graphs in quantum field theory are another example of asymptotic expansions which often do not converge.

## Use in applied mathematics

Asymptotic analysis is a key tool for exploring the ordinary and partial differential equations which arise in the mathematical modelling of real-world phenomena.[1] An illustrative example is the derivation of the boundary layer equations from the full Navier-Stokes equations governing fluid flow. In many cases, the asymptotic expansion is in power of a small parameter, $\epsilon$: in the boundary layer case, this is the nondimensional ratio of the boundary layer thickness to a typical lengthscale of the problem. Indeed, applications of asymptotic analysis in mathematical modelling often[1] centre around a nondimensional parameter which has been shown, or assumed, to be small through a consideration of the scales of the problem at hand.

## Method of dominant balance

The method of dominant balance is used to determine the asymptotic behavior of solutions to an ODE without solving it. The process is iterative in that the result obtained by performing the method once can be used as input when the method is repeated, to obtain as many terms in the asymptotic expansion as desired.

The process is as follows:

1. Assume that the asymptotic behavior has the form

$y(x) \sim e^{S(x)}\,$.

2. Make a clever guess as to which terms in the ODE may be negligible in the limit we are interested in.

3. Drop those terms and solve the resulting ODE.

4. Check that the solution is consistent with step 2. If this is the case, then we have the controlling factor of the asymptotic behavior. Otherwise, we need to try dropping different terms in step 2.

5. Repeat the process using our result as the first term of the solution.

### Example

Consider this second order ODE:

$xy''+(c-x)y'-ay=0\,$
where c and a are arbitrary constants.

This differential equation cannot be solved exactly. However, it may be useful to know how the solutions behave for large x.

We start by assuming $y\sim e^{S(x)}\,$ as $x\rightarrow \infty$. We do this with the benefit of hindsight, to make things quicker. Since we only care about the behavior of y in the large x limit, we set y equal to $e^{S(x)}\,$, and re-express the ODE in terms of S(x):

$xS''+xS'^2+(c-x)S'-a=0\,$, or
$S''+S'^2+\left(\frac{c}{x}-1\right)S'-\frac{a}{x}=0\,$
where we have used the product rule and chain rule to find the derivatives of y.

Now let us suppose that a solution to this new ODE satisfies

$S'^2\sim S'\,$ as $x\to\infty$
$S'',\frac{c}{x}S',\frac{a}{x}=o(S'^2),o(S')\,$ as $x\to\infty$

We get the dominant asymptotic behaviour by setting

$S_0'^2=S_0'\,$

If S0 satisfies the above asymptotic conditions, then everything is consistent. The terms we dropped will indeed have been negligible with respect to the ones we kept. S0 is not a solution to the ODE for S, but it represents the dominant asymptotic behaviour, which is what we are interested in. Let us check that this choice for S0 is consistent:

$S_0'=1\,$
$S_0'^2=1\,$
$S_0''=0=o(S_0')\,$
$\frac{c}{x}S_0'=\frac{c}{x}=o(S_0')\,$
$\frac{a}{x}=o(S_0')\,$

Everything is indeed consistent. Thus we find the dominant asymptotic behaviour of a solution to our ODE:

$S_0=x\,$
$y\sim e^x\,$

By convention, the asymptotic series is written as:

$y\sim Ax^p e^{\lambda x^r}\left(1+\frac{u_1}{x}+\frac{u_2}{x^2}\cdots\right)\,$

so to get at least the first term of this series we have to do another step to see if there is a power of x out the front.

We proceed by making an ansatz that we can write

$S(x)=S_0(x)+C(x)\,$

and then attempt to find asymptotic solutions for C(x). Substituting into the ODE for S(x) we find

$C''+C'^2+C'+\frac{c}{x}C'+\frac{c-a}{x}=0\,$

Repeating the same process as before, we keep C' and (c-a)/x and find that

$C_0=\log x^{a-c}\,$

The leading asymptotic behaviour is therefore

$y\sim x^{a-c}e^x\,$

## References

1. ^ a b S. Howison, Practical Applied Mathematics, Cambridge University Press, Cambridge, 2005. ISBN 0-521-60369-2
• J. P. Boyd, "The Devil's Invention: asymptotic, superasymptotic and hyperasymptotic series", Acta Applicandae Mathematicae, 56: 1-98 (1999). Preprint.
• A. Erdélyi, Asymptotic Expansions. New York: Dover, 1987.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• asymptotic analysis — noun Any of several methods for describing and analyzing the behaviour of a system at its limits …   Wiktionary

• Asymptotic theory — is the branch of mathematics which studies properties of asymptotic expansions.The most known result of this field is the prime number theorem:Let pi;( x ) be the number of prime numbers that are smaller than or equal to x .The limit:lim {x… …   Wikipedia

• Analysis of algorithms — To analyze an algorithm is to determine the amount of resources (such as time and storage) necessary to execute it. Most algorithms are designed to work with inputs of arbitrary length. Usually the efficiency or running time of an algorithm is… …   Wikipedia

• Asymptotic expansion — In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a …   Wikipedia

• Asymptotic computational complexity — In computational complexity theory, asymptotic computational complexity is the usage of the asymptotic analysis for the estimation of computational complexity of algorithms and computational problems, commonly associated with the usage of the big …   Wikipedia

• Asymptotic gain model — The asymptotic gain model [Middlebrook, RD: Design oriented analysis of feedback amplifiers ; Proc. of National Electronics Conference, Vol. XX, Oct. 1964, pp. 1 4] G {infin} T} . while in classical feedback theory, in terms of the open loop gain …   Wikipedia

• Method of matched asymptotic expansions — In mathematics, particularly in solving singularly perturbed differential equations, the method of matched asymptotic expansions is a common approach to finding an accurate approximation to a problem s solution. Contents 1 Method overview 2… …   Wikipedia

• Multiple-scale analysis — In mathematics and physics, multiple scale analysis (also called the method of multiple scales) comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large… …   Wikipedia

• List of real analysis topics — This is a list of articles that are considered real analysis topics. Contents 1 General topics 1.1 Limits 1.2 Sequences and Series 1.2.1 Summation Methods …   Wikipedia

• Clifford analysis — Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis and geometry, together with their applications. Examples of Dirac type operators include, but… …   Wikipedia