From: J F Harper Subject: Re: Theorems/Lemmas Date: Fri, 5 Mar 1999 08:51:15 +1300 (NZD) Newsgroups: [missing] To: Dave Rusin Keywords: What is Watson's Lemma (asymptotic expansions) On Thu, 4 Mar 1999, Dave Rusin wrote: > OK, I racked my brain for a full 10 seconds and came up empty. What's > Watson's Lemma? If F(x) = integral 0 to T of exp(-xt)f(t) dt, T > 0, (T may be infinite, in which case F is the Laplace transform of f), and f can be expanded in an ascending series of powers of t (Taylor or Laurent for example) then you can do the integral term by term and get a valid asymptotic series for large x in descending powers of x. See for example Carrier Krook and Pearson, Functions of a Complex Variable Theory and Technique, or Olver, Introduction to Asymptotics and Special Functions, for the precise conditions for validity, error after n terms, and how far you can extend it from real to complex x. John Harper, School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand e-mail john.harper@vuw.ac.nz phone (+64)(4)463 5341 fax (+64)(4)463 5045 ============================================================================== From: J F Harper Subject: Re: Theorems/Lemmas Date: Fri, 5 Mar 1999 09:36:31 +1300 (NZD) Newsgroups: [missing] To: Dave Rusin On Thu, 4 Mar 1999, Dave Rusin wrote: > Thanks! Any idea why it's his _lemma_? I mean, what _theorem_ did this > prep for? Presumably finding asymptotic expansions of Bessel functions of large order, or proving that Debye's method of steepest descent works. See Jeffreys & Jeffreys methods of Mathematical Physics, and Watson's Bessel Functions p236 (where of course it is just labelled Lemma: it isn't done for a book by X to call X's Lemma or X's Theorem that.) Watson's original publication, which I have not looked up, was Proc Lond Math Soc (2) (17) 1918 133. John Harper, School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand e-mail john.harper@vuw.ac.nz phone (+64)(4)463 5341 fax (+64)(4)463 5045