Asymptotic expansion

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Asymptotic expansion

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré

Henri Poincaré

Jules Henri Poincar? was a French mathematician and theoretical physicist, and a philosophy of science. Poincar? is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime....
) 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 given function as the argument of the function tends towards a particular, often infinite, point.

If φ_n is a sequence of continuous functions on some domain, and if L is a (possibly infinite) limit point of the domain, then the sequence constitutes an asymptotic scale if for every n, .

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In mathematics

Mathematics

Henri Poincaré

Asymptotic analysis

In pure mathematics and applied mathematics, particularly the analysis of algorithms, real analysis, and engineering, asymptotic analysis is a method of describing Limit ing behaviour....
and big O notation

Big O notation

In mathematics, big O notation describes the asymptotic analysis of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions....
for the notation.

The most common type of asymptotic expansion is a power series in either positive or negative terms. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace

Laplace transform

In mathematics, the Laplace transform is one of the best known and most widely used integral transforms. It is commonly used to produce an easily solvable algebraic equation from an ordinary differential equation....
and Mellin

Mellin transform

In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative group version of the two-sided Laplace transform....
transforms. Repeated integration by parts will often lead to an asymptotic expansion.

Since a convergent Taylor series

Taylor series

In mathematics, the Taylor series is a representation of a function as an Series of terms calculated from the values of its derivatives at a single point....
fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies a non-convergent series. Despite non-convergence, the asymptotic expansion is useful when truncated to a finite number of terms.

Examples of asymptotic expansions

Gamma function
Gamma function

In mathematics, the Gamma function is an extension of the factorial function to real number and complex number numbers. For a complex number z with positive real part the Gamma function is defined by...

Exponential integral
Exponential integral

In mathematics, the exponential integral is a special function defined on the complex plane given the symbol ....

Riemann zeta function
Riemann zeta function

In mathematics, the Riemann zeta function, named after Germany mathematician Bernhard Riemann, is a prominent function of great significance in number theory because of its relation to the prime number theorem....

>where are Bernoulli numbers and is a rising factorial. This expansion is valid for all complex s and is often used to compute the zeta function by using a large enough value of N, for instance .

Error function
Error function

In mathematics, the error function is a special function which occurs in probability, statistics, materials science, and partial differential equations....

Detailed 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. Thus, for example, one may 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

Exponential integral

In mathematics, the exponential integral is a special function defined on the complex plane given the symbol ....
. The integral on the right hand side, after the substitution , may be recognized as the gamma function

Gamma function

In mathematics, the Gamma function is an extension of the factorial function to real number and complex number numbers. For a complex number z with positive real part the Gamma function is defined by...
. 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 truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of for sufficiently small t. Substituting and noting that results in the asymptotic expansion given earlier in this article.