# 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)

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

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

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

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

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)