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

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

Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation.

Definition edit

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

if and only if (de Bruijn 1981, §1.4)

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


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

Properties edit

If   and  , then, under some mild conditions,[further explanation needed] the following hold:

  •  , for every real r
  •   if  

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

Examples of asymptotic formulas edit

  • Factorial
    —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.
  • Airy function
    The Airy function, Ai(x), is a solution of the differential equation y″xy = 0; it has many applications in physics.
  • Hankel functions

Asymptotic expansion edit

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   but also   and   for each fixed k. In view of the definition of the   symbol, the last equation means   in the little o notation, i.e.,   is much smaller than  

The relation   takes its full meaning if   for all k, which means the   form an asymptotic scale. In that case, some authors may abusively write   to denote the statement   One should however be careful that this is not a standard use of the   symbol, and that it does not correspond to the definition given in § Definition.

In the present situation, this relation   actually follows from combining steps k and k−1; by subtracting   from   one gets   i.e.  

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 edit

  • Gamma function
  • Exponential integral
  • Error function
    where m!! is the double factorial.

Worked example edit

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


The expression on the left is valid on the entire complex plane  , while the right hand side converges only for  . Multiplying by   and integrating both sides yields


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  , may be recognized as the gamma function. Evaluating both, one obtains the asymptotic expansion


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  . Substituting   and noting that   results in the asymptotic expansion given earlier in this article.

Asymptotic distribution edit

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 Zi 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 Zi 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   y becomes arbitrarily small in magnitude as x increases.

Applications edit

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.

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 length scale 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.

Asymptotic versus Numerical Analysis edit

Debruijn illustrates the use of asymptotics in the following dialog between Miss N.A., a Numerical Analyst, and Dr. A.A., an Asymptotic Analyst:

N.A.: I want to evaluate my function   for large values of  , with a relative error of at most 1%.

A.A.:  .

N.A.: I am sorry, I don't understand.


N.A.: But my value of   is only 100.

A.A.: Why did you not say so? My evaluations give


N.A.: This is no news to me. I know already that  .

A.A.: I can gain a little on some of my estimates. Now I find that


N.A.: I asked for 1%, not for 20%.

A.A.: It is almost the best thing I possibly can get. Why don't you take larger values of  ?

N.A.: !!! I think it's better to ask my electronic computing machine.

Machine: f(100) = 0.01137 42259 34008 67153

A.A.: Haven't I told you so? My estimate of 20% was not far off from the 14% of the real error.

N.A.: !!! . . .  !

Some days later, Miss N.A. wants to know the value of f(1000), but her machine would take a month of computation to give the answer. She returns to her Asymptotic Colleague, and gets a fully satisfactory reply.[4]

See also edit

Notes edit

  1. ^ "Asymptotic equality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  2. ^ Estrada & Kanwal (2002, §1.2)
  3. ^ a b Howison, S. (2005), Practical Applied Mathematics, Cambridge University Press
  4. ^ Bruijn, Nicolaas Govert de (1981). Asymptotic methods in analysis. Dover books on advanced mathematics. New York: Dover publ. p. 19. ISBN 978-0-486-64221-5.

References edit

External links edit