# Asymptotic expansion

of a function

A series

such that for any integer one has

 (1)

where is some given asymptotic sequence as . In such a case one also has

 (2)

The sequence is omitted from formula (2) if it is clear from the context which sequence is meant.

The asymptotic expansion (2) is called an asymptotic expansion in the sense of Erdélyi [3]. An expansion of the type

 (3)

where are constants, is called an asymptotic expansion in the sense of Poincaré. If the asymptotic sequence of functions is given, the asymptotic expansion (3), contrary to the expansion (2), is uniquely defined by the function itself. If (1) is valid for a finite number of values , then (1) is called an asymptotic expansion up to . The series

are known as asymptotic series. As a rule such series are divergent. Asymptotic power series are the ones most commonly employed; the corresponding asymptotic expansions are asymptotic expansions in the sense of Poincaré.

The following is an example of an asymptotic expansion in the sense of Erdélyi:

, where is the Bessel function, and

The concepts of an asymptotic expansion of a function and of an asymptotic series were introduced by H. Poincaré [1] in the context of problems in celestial mechanics. Special cases of asymptotic expansions were discovered and utilized as early as the 18th century [2]. Asymptotic expansions play an important role in many problems in mathematics, mechanics and physics. This is because many problems do not admit exact solutions, but their solutions can be obtained as asymptotic approximations. Moreover, numerical methods are often disregarded if asymptotic approximations can be relatively easily found.

#### References

 [1] H. Poincaré, "Sur les intégrales irrégulières des équations linéaires" Acta Math. , 8 (1886) pp. 295–344 [2] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 2 [3] A. Erdélyi, M. Wyman, "The asymptotic evaluation of certain integrals" Arch. Rational Mech. Anal. , 14 (1963) pp. 217–260