Asymptotic expansion

of a function $ f(x) $

A series

$$ \sum _ {n=0 } ^ \infty \psi _ {n} (x) $$

such that for any integer $ N \geq 0 $ one has

$$ \tag{1 } f (x) = \sum _ {n=0 } ^ { N } \psi _ {n} (x) + o ( \phi _ {N} (x) ) \ \ (x \rightarrow x _ {0} ), $$

where $ \{ \phi _ {n} (x) \} $ is some given asymptotic sequence as $ x \rightarrow x _ {0} $. In such a case one also has

$$ \tag{2 } f (x) \sim \sum _ {n=0 } ^ \infty \psi _ {n} (x),\ \ \{ \phi _ {n} (x) \} ,\ \ ( x \rightarrow x _ {0} ). $$

The sequence $ \{ \phi _ {n} (x) \} $ 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

$$ \tag{3 } f (x) \sim \sum _ {n=0 } ^ \infty a _ {n} \phi _ {n} (x) \ \ ( x \rightarrow x _ {0} ), $$

where $ a _ {n} $ are constants, is called an asymptotic expansion in the sense of Poincaré. If the asymptotic sequence of functions $ \{ \phi _ {n} (x) \} $ is given, the asymptotic expansion (3), contrary to the expansion (2), is uniquely defined by the function $ f(x) $ itself. If (1) is valid for a finite number of values $ N = 0 \dots N _ {0} < \infty $, then (1) is called an asymptotic expansion up to $ o( \phi _ {N _ {0} } (x)) $. The series

$$ \sum _ {n=0 } ^ \infty \psi _ {n} (x),\ \ \sum _ {n=0 } ^ \infty a _ {n} \phi _ {n} (x) $$

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:

$$ J _ \nu (x) \sim \ \sqrt { \frac{2}{\pi x } } \left [ \cos \left ( x - \frac{\pi \nu }{2} - \frac \pi {4} \right ) \sum _ {n=0 } ^ \infty ( -1 ) ^ {n} a _ {2n} x ^ {-2n}\right. - $$

$$ - \left . \sin \left ( x - \frac{\pi \nu }{2} - \frac{\pi}{4} \right ) \sum _ {n=0 } ^ \infty ( -1 ) ^ {n} a _ {2n+1} x ^ {-2n-1} \right ] $$

$ (x \rightarrow + \infty ) $, where $ J _ \nu (x) $ is the Bessel function, and

$$ a _ {n} = \frac{\Gamma ( \nu + n + 1 / 2 ) }{2 ^ {n} n! \Gamma ( \nu - n + 1 / 2 ) } . $$

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
[a1]	N.G. de Bruijn, "Asymptotic methods in analysis" , Dover, reprint (1981)
[a2]	A. Erdélyi, "Asymptotic expansions" , Dover, reprint (1956)
[a3]	E.T. Copson, "Asymptotic expansions" , Cambridge Univ. Press (1965)
[a4]	J.P. Murray, "Asymptotic analysis" , Springer (1984)
[a5]	N. Bleistein, R.A. Handelsman, "Asymptotic expansions of integrals" , Dover, reprint (1986) pp. Chapts. 1, 3, 5