In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, asymptotic analysis, also known as asymptotics, is a method of describing
limiting
In electronics, a limiter is a circuit that allows signals below a specified input power or level to pass unaffected while attenuating (lowering) the peaks of stronger signals that exceed this threshold. Limiting is a type of dynamic range comp ...
behavior.
As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as becomes very large, the term becomes insignificant compared to . The function is said to be "''asymptotically equivalent'' to , as ". This is often written symbolically as , which is read as " is asymptotic to ".
An example of an important asymptotic result is the
prime number theorem. Let denote the
prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ).
History
Of great interest in number theory is t ...
(which is not directly related to the constant
pi), i.e. is the number of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s that are less than or equal to . Then the theorem states that
Asymptotic analysis is commonly used in
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
as part of the
analysis of algorithms
In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that re ...
and is often expressed there in terms of
big O notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
.
Definition
Formally, given functions and , we define a binary relation
if and only if
The symbol is the
tilde
The tilde () or , is a grapheme with several uses. The name of the character came into English from Spanish, which in turn came from the Latin '' titulus'', meaning "title" or "superscription". Its primary use is as a diacritic (accent) in ...
. The relation is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
on the set of functions of ; the functions and are said to be ''asymptotically equivalent''. The
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
of and 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. , , . 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 is zero infinitely often as goes to the limiting value. For that reason, some authors use an alternative definition. The alternative definition, in
little-o notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Land ...
, is that if and only if
This definition is equivalent to the prior definition if is not zero in some
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
of the limiting value.
Properties
If
and
, as
, then the following hold:
*
, for every real
*
if
*
*
Such properties allow asymptotically-equivalent functions to be freely exchanged in many algebraic expressions.
Note that those properties are only correct if and only if
tends to infinity (in other words, those properties are only applied for sufficiently large value of
). If
does not tend to infinity, but instead to some arbitrary finite constants
, then the following limit from above definition:
≠ 1 , for some constant
Similarly:
≠ 1 , for some constant
Thus, those respective functions are no longer asymptotically-equivalent and cannot be applied above properties.
A simple example for this, let
and
, we can see that:
However:
Hence,
and
are not asymptotically-equivalent as
.
Examples of asymptotic formulas
*
Factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
—this is
Stirling's approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less p ...
*
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
In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function , are linearly independent solut ...
The Airy function, Ai(''x''), is a solution of the differential equation ; it has many applications in physics.
*
Hankel functions
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
Asymptotic expansion
An
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 ...
of a
Finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
is in practice an expression of that function in terms of a
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
, the
partial sum
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
s of which do not necessarily converge, but such that taking any initial partial sum provides an asymptotic formula for . The idea is that successive terms provide an increasingly accurate description of the order of growth of .
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
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
, i.e.,
is much smaller than
The relation
takes its full meaning if
for all ''k'', which means the
form an
asymptotic scale 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 ...
. 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 .
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
*
Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
*
Exponential integral
In mathematics, the exponential integral Ei is a special function on the complex plane.
It is defined as one particular definite integral of the ratio between an exponential function and its argument.
Definitions
For real non-zero values of  ...
*
Error function
In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as:
:\operatorname z = \frac\int_0^z e^\,\mathrm dt.
This integral is a special (non-elementary ...
where is the
double factorial
In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is,
:n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots.
For even , the ...
.
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
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
In mathematics, the exponential integral Ei is a special function on the complex plane.
It is defined as one particular definite integral of the ratio between an exponential function and its argument.
Definitions
For real non-zero values of  ...
. The integral on the right hand side, after the substitution
, may be recognized as the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
. 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
In
mathematical statistics
Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical an ...
, an
asymptotic distribution
In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the "limiting" distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing ...
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 for , for some positive integer . An asymptotic distribution allows to range without bound, that is, is infinite.
A special case of an asymptotic distribution is when the late entries go to zero—that is, the go to 0 as goes to infinity. Some instances of "asymptotic distribution" refer only to this special case.
This is based on the notion of an
asymptotic
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
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
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
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
Asymptotic analysis is used in several
mathematical sciences
The mathematical sciences are a group of areas of study that includes, in addition to mathematics, those academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper.
Statisti ...
. In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, asymptotic theory provides limiting approximations of the
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
of
sample statistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypo ...
s, such as the
likelihood ratio
The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood functi ...
statistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypo ...
and the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
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
In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characteri ...
.
Examples of applications are the following.
* In
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
, asymptotic analysis is used to build
numerical method
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.
Mathem ...
s to approximate
equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
solutions.
* In
mathematical statistics
Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical an ...
and
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, asymptotics are used in analysis of long-run or large-sample behaviour of random variables and estimators.
* In
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
in the
analysis of algorithms
In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that re ...
, considering the performance of algorithms.
* The behavior of
physical system
A physical system is a collection of physical objects.
In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...
s, an example being
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
.
* In
accident analysis
Accident analysis is carried out in order to determine the cause or causes of an accident (that can result in single or multiple outcomes) so as to prevent further accidents of a similar kind. It is part of ''accident investigation or incident inv ...
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
Partial may refer to:
Mathematics
* Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant
** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial ...
differential equations which arise in the
mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
ling of real-world phenomena.
[Howison, S. (2005), ]
Practical Applied Mathematics
', Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
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, : 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
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
In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form
:\int_a^b e^ \, dx,
where f(x) is a twice-differentiable function, ''M'' is a large number, and the endpoints ''a'' an ...
,
saddle-point method
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point ( saddle point), in ...
,
method of steepest descent
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in r ...
) or in the approximation of probability distributions (
Edgeworth series The Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution in terms of its cumulants. The se ...
). The
Feynman graphs in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
are another example of asymptotic expansions which often do not converge.
See also
*
Asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
*
Asymptotic computational complexity In computational complexity theory, asymptotic computational complexity is the usage of asymptotic analysis for the estimation of computational complexity of algorithms and computational problems, commonly associated with the usage of the big O no ...
*
Asymptotic density In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the des ...
(in number theory)
*
Asymptotic theory (statistics) In statistics, asymptotic theory, or large sample theory, is a framework for assessing properties of estimators and statistical tests. Within this framework, it is often assumed that the sample size may grow indefinitely; the properties of estimato ...
*
Asymptotology
*
Big O notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
*
Leading-order term The leading-order terms (or corrections) within a mathematical equation, expression or model are the terms with the largest order of magnitude.J.K.Hunter, ''Asymptotic Analysis and Singular Perturbation Theory'', 2004. http://www.math.ucdavis.edu ...
*
Method of dominant balance In mathematics, the method of dominant balance is used to determine the asymptotic behavior of solutions to an ordinary differential equation without fully solving the equation. The process is iterative, in that the result obtained by performing the ...
(for ODEs)
*
Method of matched asymptotic expansions In mathematics, the method of matched asymptotic expansions is a common approach to finding an accurate approximation to the solution to an equation, or system of equations. It is particularly used when solving singularly perturbed differential equ ...
*
Watson's lemma
Notes
References
*
*
*
*
*
* {{citation, last1=Paris, first1= R. B., last2= Kaminsky, first2= D. , year=2001, title= Asymptotics and Mellin-Barnes Integrals, publisher=
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
, url=https://www.researchgate.net/publication/39064661
External links
''Asymptotic Analysis'' —home page of the journal, which is published by
IOS Press
IOS Press is a publishing house headquartered in Amsterdam, specialising in the publication of journals and books related to fields of scientific, technical, and medical research. Established in 1987, IOS Press publishes around 100 internationa ...
A paper on time series analysis using asymptotic distribution
Mathematical series