# Eikonal Blog

## 2010.01.04

### Asymptotic expansion

Filed under: mathematics — Tags: , — sandokan65 @ 20:22

Definition:

• a series $\sum_{k=0}^\infty \frac{a_k}{z^k}$ is \underline{an asymptotic expansion} of $f(z)$ iff
• $\lim_{|z|\rightarrow\infty} z^n\{f(z)-\sum_{k=0}^n \frac{a_k}{z^k} \} =0$ for all $n\in{\Bbb N}_0$.

Note:

• The asymptotic expansions are not unique. For example $f(z)$ and $F(z)+e^{-z}$ have the same asymptotic expansion at $z\rightarrow +\infty$.

Watson Lemma:

• If $F(\tau)$ has following expansion near $\tau =0$: $F(\tau) = \sum_{n=1}^\infty a_n \tau^{\frac{n}{r}-1}$ (for $|\tau| \le a$ and $r>0$),
• and if there exist $K$ and $b$ s/t $|F(\tau)| < K e^{b|\tau|}$ for $|\tau| \ge a$,
• then, for large $|\nu|$ (and for $|\arg \nu| < \frac\pi2$):
• ${\cal F}(\nu) :\equiv \int_0^\infty e^{-\nu \tau} F(\tau) d\tau \sim \sum_{n=1}^\infty a_n \Gamma(\frac{n}{\nu}) \nu^{-\frac{n}{r}}$.

Source:

• “The Functions of Mathematical Physics” by Harry Hochstadt; Ch3. The Gamma Function