Asymptotic expansion

Summary

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. Investigations by Dingle (1973) revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function.

The most common type of asymptotic expansion is a power series in either positive or negative powers. 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.

Since a convergent Taylor series 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. The approximation may provide benefits by being more mathematically tractable than the function being expanded, or by an increase in the speed of computation of the expanded function. Typically, the best approximation is given when the series is truncated at the smallest term. This way of optimally truncating an asymptotic expansion is known as superasymptotics.[1] The error is then typically of the form ~ exp(−c/ε) where ε is the expansion parameter. The error is thus beyond all orders in the expansion parameter. It is possible to improve on the superasymptotic error, e.g. by employing resummation methods such as Borel resummation to the divergent tail. Such methods are often referred to as hyperasymptotic approximations.

See asymptotic analysis and big O notation for the notation used in this article.

Formal definition edit

First we define an asymptotic scale, and then give the formal definition of an asymptotic expansion.

If   is a sequence of continuous functions on some domain, and if   is a limit point of the domain, then the sequence constitutes an asymptotic scale if for every n,

 

(  may be taken to be infinity.) In other words, a sequence of functions is an asymptotic scale if each function in the sequence grows strictly slower (in the limit  ) than the preceding function.

If   is a continuous function on the domain of the asymptotic scale, then f has an asymptotic expansion of order   with respect to the scale as a formal series

 

if

 

or the weaker condition

 

is satisfied. If one or the other holds for all  , then we write[citation needed]

 

In contrast to a convergent series for  , wherein the series converges for any fixed   in the limit  , one can think of the asymptotic series as converging for fixed   in the limit   (with   possibly infinite).

Examples edit

 
Plots of the absolute value of the fractional error in the asymptotic expansion of the Gamma function (left). The horizontal axis is the number of terms in the asymptotic expansion. Blue points are for x = 2 and red points are for x = 3. It can be seen that the least error is encountered when there are 14 terms for x = 2, and 20 terms for x = 3, beyond which the error diverges.
  • Gamma function (Stirling's approximation)
     
  • Exponential integral
     
  • Logarithmic integral
     
  • Riemann zeta function
     
    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
     
    where (2n − 1)!! 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. 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

 

after the substitution   on the right hand side. The integral on the left hand side, understood as a Cauchy principal value, can be expressed in terms of the exponential integral. The integral on the right hand side 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 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.

Integration by parts edit

Using integration by parts, we can obtain an explicit formula[2]

 
For any fixed  , the absolute value of the error term   decreases, then increases. The minimum occurs at  , at which point  . This bound is said to be "asymptotics beyond all orders".

Properties edit

Uniqueness for a given asymptotic scale edit

For a given asymptotic scale   the asymptotic expansion of function   is unique.[3] That is the coefficients   are uniquely determined in the following way:

 
where   is the limit point of this asymptotic expansion (may be  ).

Non-uniqueness for a given function edit

A given function   may have many asymptotic expansions (each with a different asymptotic scale).[3]

Subdominance edit

An asymptotic expansion may be an asymptotic expansion to more than one function.[3]

See also edit

Related fields edit

Asymptotic methods edit

Notes edit

  1. ^ Boyd, John P. (1999), "The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series" (PDF), Acta Applicandae Mathematicae, 56 (1): 1–98, doi:10.1023/A:1006145903624, hdl:2027.42/41670.
  2. ^ O’Malley, Robert E. (2014), O'Malley, Robert E. (ed.), "Asymptotic Approximations", Historical Developments in Singular Perturbations, Cham: Springer International Publishing, pp. 27–51, doi:10.1007/978-3-319-11924-3_2, ISBN 978-3-319-11924-3, retrieved 2023-05-04
  3. ^ a b c S.J.A. Malham, "An introduction to asymptotic analysis", Heriot-Watt University.

References edit

External links edit