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 given function as the argument of the function tends towards a particular, often infinite, point. For other meanings of mathematics or math, see mathematics (disambiguation). ... Jules Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [][1]), was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ...

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, . If f is a continuous function on the domain of the asymptotic scale, then an asymptotic expansion of f with respect to the scale is a formal series such that, for any fixed N, In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion 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. ...

In this case, we write

.

See asymptotic analysis and big O notation for the notation. In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ... It has been suggested that this article or section be merged into Asymptotic notation. ...

The most common type of asymptotic expansion is a power series in either positive or negative terms. While a convergent Taylor series fits the definition as given, a non-convergent series is what is usually intended by the phrase. Methods of generating such expansions include the Euler-Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion. As the degree of the Taylor series rises, it approaches the correct function. ... In mathematics, the Euler-Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. ... In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ... In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. ...

Examples of asymptotic expansions

• Gamma function

• Exponential integral

• Riemann zeta function

where B_2m 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 N > | s | .

• Error function

The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ... In mathematics, the exponential integral Ei(x) is defined as Since 1/t diverges at t=0, the above integral has to be understood in terms of the Cauchy principal value. ... In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ... In mathematics, the Pochhammer symbol is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial The empty product (x)0 is defined to be 1 in both cases. ... In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics 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 | w | < 1. Multiplying by e ^{− w / t} 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 u = w / t, may be recognized as the gamma function. Evaluating both, one obtains the asymptotic expansion In mathematics, the exponential integral Ei(x) is defined as Since 1/t diverges at t=0, the above integral has to be understood in terms of the Cauchy principal value. ... The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...

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 x = 1 / t and noting that results in the asymptotic expansion given earlier in this article.

References

• Hardy, G. H., Divergent Series, Oxford University Press, 1949
• Paris, R. B. and Kaminsky, D., Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, 2001
• Whittaker, E. and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1963