Editing Asymptotic expansion (section)

{{Short description|Series of functions in mathematics}}
In [[mathematics]], an '''asymptotic expansion''', '''asymptotic series''' or '''Poincaré expansion''' (after [[Henri Poincaré]]) is a [[formal series]] of functions which has the property that [[truncation|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 {{Harvtxt|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 theory of asymptotic series was created by Poincaré (and independently by [[Thomas Joannes Stieltjes|Stieltjes]]) in 1886.<ref>{{Cite book |last=Jahnke |first=Hans Niels |title=A history of analysis |date=2003 |publisher=American mathematical society |isbn=978-0-8218-2623-2 |series=History of mathematics |location=Providence (R.I.) |pages=190}}</ref>

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 transform|Laplace]] and [[Mellin transform|Mellin]] transforms. Repeated [[integration by parts]] will often lead to an asymptotic expansion.

Since a ''[[Convergence (mathematics)|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'''.<ref>{{citation|first=John P.|last= Boyd|title= The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series |journal= [[Acta Applicandae Mathematicae]] |volume=56|issue=1|pages=1–98| year=1999| doi= 10.1023/A:1006145903624|url=https://deepblue.lib.umich.edu/bitstream/2027.42/41670/1/10440_2004_Article_193995.pdf|hdl=2027.42/41670|hdl-access=free}}.</ref> The error is then typically of the form {{math|~&thinsp;exp(−''c''/ε)}} where {{math|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.