Asymptotic power series

An asymptotic series with respect to the sequence

$$ \{ x ^ {-n} \} \ ( x \rightarrow \infty ) $$

or with respect to a sequence

$$ \{ ( x - x _ {0} ) ^ {n} \} \ ( x \rightarrow x _ {0} ) $$

(cf. Asymptotic expansion of a function). Asymptotic power series may be added, multiplied, divided and integrated just like convergent power series.

Let two functions $ f(x) $ and $ g(x) $ have the following asymptotic expansions as $ x \rightarrow \infty $:

$$ f (x) \sim \sum _ {n = 0 } ^ \infty \frac{a _ {n} }{x ^ {n} } ,\ \ g (x) \sim \sum _ {n = 0 } ^ \infty \frac{b _ {n} }{x^n}. $$

Then

1)

$$ Af (x) + Bg (x) \sim \sum _ {n =0} ^ \infty \frac{A a _ {n} + B b _ {n} }{x^n} $$

($A, B $ are constants);

2)

$$ f (x) g (x) \sim \sum _ {n = 0} ^ \infty \frac{c _ {n} }{x^n } ; $$

3)

$$ \frac{1}{f(x)} \sim \frac{1}{a_0} + \sum _ {n = 1 } ^ \infty \frac{d_n} {x^n} ,\ a _ {0} \neq 0 $$

($c _ {n} , d _ {n} $ are calculated as for convergent power series);

4) if the function $ f(x) $ is continuous for $ x > a > 0 $, then

$$ \int\limits _ { x } ^ \infty \left ( f (t) - a _ {0} - \frac{a _ {1} }{t} \right ) dt \sim \sum _ {n = 1 } ^ \infty \frac{a _ {n+1} }{n x ^ {n} } ; $$

5) an asymptotic power series cannot always be differentiated, but if $ f(x) $ has a continuous derivative which can be expanded into an asymptotic power series, then

$$ f ^ { \prime } (x) \sim - \sum _ {n = 2 } ^ \infty \frac{( n - 1 ) a _ {n-1} }{x^n} . $$

Examples of asymptotic power series.

$$ \int\limits _ { x } ^ \infty \frac{e ^ {x-t} }{t} dt \sim \ \sum _ {n = 1 } ^ \infty \frac{( -1 ) ^ {n-1} ( n - 1 ) ! }{x^n} ; $$

$$ \sqrt {x } e ^ {-ix} H _ {0} ^ {(1)} (x) \sim \sum _ {n = 0 } ^ \infty \frac{e ^ {-i \pi / 4 } ( - i ) ^ {n} [ \Gamma ( n + 1 / 2 ) ] ^ {2} }{2 ^ {n-1/2} \pi ^ {3/2} n ! x ^ {n} } , $$

where $ {H _ {0} ^ {(1)} } (x) $ is the Hankel function of order zero (cf. Hankel functions) (the above asymptotic power series diverge for all $ x $).

Similar assertions are also valid for functions of a complex variable $ z $ as $ z \rightarrow \infty $ in a neighbourhood of the point at infinity or inside an angle. For a complex variable 5) takes the following form: If the function $ f(z) $ is regular in the domain $ D= \{ | z | > a, \alpha < | { \mathop{\rm arg} } z | < \beta \} $ and if

$$ f (z) \sim \sum _ {n = 0 } ^ \infty \frac{a _ {n} }{z^n} $$

uniformly in $ { \mathop{\rm arg} } z $ as $ | z | \rightarrow \infty $ inside any closed angle contained in $ D $, then

$$ f ^ { \prime } (z) \sim - \sum _ {n = 1 } ^ \infty \frac{na _ {n} }{z ^ {n+1} } $$

uniformly in $ \mathop{\rm arg} z $ as $ | z | \rightarrow \infty $ in any closed angle contained in D.

References

[1]	E.T. Copson, "Asymptotic expansions" , Cambridge Univ. Press (1965)
[2]	A. Erdélyi, "Asymptotic expansions" , Dover, reprint (1956)
[3]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952)
[a1]	N.G. de Bruijn, "Asymptotic methods in analysis" , Dover, reprint (1981)