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
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