Asymptotic analysis

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.

This article is about the behavior of functions as inputs approach infinity, or some other limit value. For asymptotes in geometry, see Asymptote.

As an illustration, suppose that we are interested in the properties of a function f(n) as n becomes very large. If f(n) = n² + 3n, then as n becomes very large, the term 3n becomes insignificant compared to n². The function f(n) is said to be "asymptotically equivalent to n², as n → ∞". This is often written symbolically as f(n) ~ n², which is read as "f(n) is asymptotic to n²".

An example of an important asymptotic result is the prime number theorem. Let π(x) denote the prime-counting function (which is not directly related to the constant pi), i.e. π(x) is the number of prime numbers that are less than or equal to x. Then the theorem states that

${\displaystyle \pi (x)\sim {\frac {x}{\ln x}}.}$

Definition

Formally, given functions f(x) and g(x), we define a binary relation

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

if and only if (de Bruijn 1981, §1.4)

${\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=1.}$

The symbol ~ is the tilde. The relation is an equivalence relation on the set of functions of x; the functions f and g are said to be asymptotically equivalent. The domain of f and g can be any set for which the limit is defined: e.g. real numbers, complex numbers, positive integers.

The same notation is also used for other ways of passing to a limit: e.g. x → 0, x ↓ 0, |x| → 0. The way of passing to the limit is often not stated explicitly, if it is clear from the context.

Although the above definition is common in the literature, it is problematic if g(x) is zero infinitely often as x goes to the limiting value. For that reason, some authors use an alternative definition. The alternative definition, in little-o notation, is that f ~ g if and only if

${\displaystyle f(x)-g(x)=o(g(x)).}$

This definition is equivalent to the prior definition if g(x) is not zero in some neighbourhood of the limiting value.[1][2]

Properties

If ${\displaystyle f\sim g}$ and ${\displaystyle a\sim b}$, then under some mild conditions, the following hold.

• ${\displaystyle f^{r}\sim g^{r}}$, for every real r
• ${\displaystyle \log(f)\sim \log(g)}$
• ${\displaystyle f\times a\sim g\times b}$
• ${\displaystyle f/a\sim g/b}$

Such properties allow asymptotically-equivalent functions to be freely exchanged in many algebraic expressions.

Examples of asymptotic formulas

• Factorial

${\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}$

—this is Stirling's approximation

• Partition function

For a positive integer n, the partition function, p(n), gives the number of ways of writing the integer n as a sum of positive integers, where the order of addends is not considered.

${\displaystyle p(n)\sim {\frac {1}{4n{\sqrt {3}}}}e^{\pi {\sqrt {\frac {2n}{3}}}}}$

• Airy function

The Airy function, Ai(x), is a solution of the differential equation  y'' − xy = 0; it has many applications in physics.

${\displaystyle \operatorname {Ai} (x)\sim {\frac {e^{-{\frac {2}{3}}x^{\frac {3}{2}}}}{2{\sqrt {\pi }}x^{1/4}}}}$

• Hankel functions

${\displaystyle {\begin{aligned}H_{\alpha }^{(1)}(z)&\sim {\sqrt {\frac {2}{\pi z}}}e^{i\left(z-{\frac {2\pi \alpha -\pi }{4}}\right)}\\H_{\alpha }^{(2)}(z)&\sim {\sqrt {\frac {2}{\pi z}}}e^{-i\left(z-{\frac {2\pi \alpha -\pi }{4}}\right)}\end{aligned}}}$

Construction

General

Consider:

${\displaystyle h(x)=f(x)(1-F(x))+g(x)F(x)}$

where ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$ are real-valued analytic functions, and ${\displaystyle F(x)}$ is a Cumulative distribution function.

Then ${\displaystyle h(x)}$ is asymptotic to ${\displaystyle f(x)}$ as ${\displaystyle x\to (-\infty )}$ and asymptotic to ${\displaystyle g(x)}$ as ${\displaystyle x\to (+\infty )}$.

Asymptotic to two different polynomials

Suppose we want a real-valued function that is asymptotic to ${\displaystyle (a_{0}+a_{1}x)}$ as ${\displaystyle x\to (-\infty )}$ and is asymptotic to ${\displaystyle (b_{0}+b_{1}x)}$ as ${\displaystyle x\to (+\infty )}$. Then

${\displaystyle h(x)=(a_{0}+a_{1}x)(1-F(x))+(b_{0}+b_{1}x)F(x)}$

will do this.

Asymptotic expansion

Main article: 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 an increasingly accurate description of the order of growth of f.

In symbols, it means we have ${\displaystyle f\sim g_{1},}$ but also ${\displaystyle f-g_{1}\sim g_{2}}$ and ${\displaystyle f-g_{1}-\cdots -g_{k-1}\sim g_{k}}$ for each fixed k. In view of the definition of the ${\displaystyle \sim }$ symbol, the last equation means ${\displaystyle f-(g_{1}+\cdots +g_{k})=o(g_{k})}$ in the little o notation, i.e., ${\displaystyle f-(g_{1}+\cdots +g_{k})}$ is much smaller than ${\displaystyle g_{k}.}$

The relation ${\displaystyle f-g_{1}-\cdots -g_{k-1}\sim g_{k}}$ takes its full meaning if ${\displaystyle g_{k+1}=o(g_{k})}$ for all k, which means the ${\displaystyle g_{k}}$ form an asymptotic scale. In that case, some authors may abusively write ${\displaystyle f\sim g_{1}+\cdots +g_{k}}$ to denote the statement ${\displaystyle f-(g_{1}+\cdots +g_{k})=o(g_{k}).}$ One should however be careful that this is not a standard use of the ${\displaystyle \sim }$ symbol, and that it does not correspond to the definition given in § Definition.

In the present situation, this relation ${\displaystyle g_{k}=o(g_{k-1})}$ actually follows from combining steps k and k−1; by subtracting ${\displaystyle f-g_{1}-\cdots -g_{k-2}=g_{k-1}+o(g_{k-1})}$ from ${\displaystyle f-g_{1}-\cdots -g_{k-2}-g_{k-1}=g_{k}+o(g_{k}),}$ one gets ${\displaystyle g_{k}+o(g_{k})=o(g_{k-1}),}$ i.e. ${\displaystyle g_{k}=o(g_{k-1}).}$

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. This optimal partial sum will usually have more terms as the argument approaches the limit value.

Examples of asymptotic expansions

• Gamma function

${\displaystyle {\frac {e^{x}}{x^{x}{\sqrt {2\pi x}}}}\Gamma (x+1)\sim 1+{\frac {1}{12x}}+{\frac {1}{288x^{2}}}-{\frac {139}{51840x^{3}}}-\cdots \ (x\to \infty )}$

• Exponential integral

${\displaystyle xe^{x}E_{1}(x)\sim \sum _{n=0}^{\infty }{\frac {(-1)^{n}n!}{x^{n}}}\ (x\to \infty )}$

• Error function

${\displaystyle {\sqrt {\pi }}xe^{x^{2}}{\rm {erfc}}(x)\sim 1+\sum _{n=1}^{\infty }(-1)^{n}{\frac {(2n-1)!!}{n!(2x^{2})^{n}}}\ (x\to \infty )}$

where (2n − 1)!! is the double factorial.

Worked example

Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. For example, we might start with the ordinary series

${\displaystyle {\frac {1}{1-w}}=\sum _{n=0}^{\infty }w^{n}}$

The expression on the left is valid on the entire complex plane ${\displaystyle w\neq 1}$, while the right hand side converges only for ${\displaystyle |w|<1}$. Multiplying by ${\displaystyle e^{-w/t}}$ and integrating both sides yields

${\displaystyle \int _{0}^{\infty }{\frac {e^{-{\frac {w}{t}}}}{1-w}}\,dw=\sum _{n=0}^{\infty }t^{n+1}\int _{0}^{\infty }e^{-u}u^{n}\,du}$

The integral on the left hand side can be expressed in terms of the exponential integral. The integral on the right hand side, after the substitution ${\displaystyle u=w/t}$, may be recognized as the gamma function. Evaluating both, one obtains the asymptotic expansion

${\displaystyle e^{-{\frac {1}{t}}}\operatorname {Ei} \left({\frac {1}{t}}\right)=\sum _{n=0}^{\infty }n!\;t^{n+1}}$

Here, the right hand side is clearly not convergent for any non-zero value of t. However, by keeping t small, and truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of ${\displaystyle \operatorname {Ei} (1/t)}$. Substituting ${\displaystyle x=-1/t}$ and noting that ${\displaystyle \operatorname {Ei} (x)=-E_{1}(-x)}$ results in the asymptotic expansion given earlier in this article.

Asymptotic distribution

Main article: Asymptotic distribution

In mathematical statistics, an asymptotic distribution is a hypothetical distribution that is in a sense the "limiting" distribution of a sequence of distributions. A distribution is an ordered set of random variables Z_i for i = 1, ..., n, for some positive integer n. An asymptotic distribution allows i to range without bound, that is, n is infinite.

A special case of an asymptotic distribution is when the late entries go to zero—that is, the Z_i go to 0 as i goes to infinity. Some instances of "asymptotic distribution" refer only to this special case.

This is based on the notion of an asymptotic function which cleanly approaches a constant value (the asymptote) as the independent variable goes to infinity; "clean" in this sense meaning that for any desired closeness epsilon there is some value of the independent variable after which the function never differs from the constant by more than epsilon.

An asymptote is a straight line that a curve approaches but never meets or crosses. Informally, one may speak of the curve meeting the asymptote "at infinity" although this is not a precise definition. In the equation ${\displaystyle y={\frac {1}{x}},}$ y becomes arbitrarily small in magnitude as x increases.

Applications

Asymptotic analysis is used in several mathematical sciences. In statistics, asymptotic theory provides limiting approximations of the probability distribution of sample statistics, such as the likelihood ratio statistic and the expected value of the deviance. Asymptotic theory does not provide a method of evaluating the finite-sample distributions of sample statistics, however. Non-asymptotic bounds are provided by methods of approximation theory.

Examples of applications are the following.

• In applied mathematics, asymptotic analysis is used to build numerical methods to approximate equation solutions.
• In mathematical statistics and probability theory, asymptotics are used in analysis of long-run or large-sample behaviour of random variables and estimators.
• in computer science in the analysis of algorithms, considering the performance of algorithms.
• the behavior of physical systems, an example being statistical mechanics.
• in accident analysis when identifying the causation of crash through count modeling with large number of crash counts in a given time and space.

Asymptotic analysis is a key tool for exploring the ordinary and partial differential equations which arise in the mathematical modelling of real-world phenomena.[3] 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, ε: 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[3] center around a nondimensional parameter which has been shown, or assumed, to be small through a consideration of the scales of the problem at hand.

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 Feynman graphs in quantum field theory are another example of asymptotic expansions which often do not converge.

See also

• Asymptote
• Asymptotic computational complexity
• Asymptotic density (in number theory)
• Asymptotic theory (statistics)
• Asymptotology
• Big O notation
• Leading-order term
• Method of dominant balance (for ODEs)
• Method of matched asymptotic expansions
• Watson's lemma

Notes

1. "Asymptotic equality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
2. Estrada & Kanwal (2002, §1.2)
3. Howison, S. (2005), Practical Applied Mathematics, Cambridge University Press

References

• Balser, W. (1994), From Divergent Power Series To Analytic Functions, Springer-Verlag, ISBN 9783540485940
• de Bruijn, N. G. (1981), Asymptotic Methods in Analysis, Dover Publications, ISBN 9780486642215
• Estrada, R.; Kanwal, R. P. (2002), A Distributional Approach to Asymptotics, Birkhäuser, ISBN 9780817681302
• Miller, P. D. (2006), Applied Asymptotic Analysis, American Mathematical Society, ISBN 9780821840788
• Murray, J. D. (1984), Asymptotic Analysis, Springer, ISBN 9781461211228
• Paris, R. B.; Kaminsky, D. (2001), Asymptotics and Mellin-Barnes Integrals, Cambridge University Press

External links

• Asymptotic Analysis  —home page of the journal, which is published by IOS Press
• A paper on time series analysis using asymptotic distribution