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  • Task info Open for submissions from Wednesday, 27 July 2016, 9:45 PM (3377 days ago)
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 Example 3, p. 265


An integral representation of the modified Bessel function \fs2 K_0(x) is
\fs2K_0(x) = e^{-x} \, \int_0^\infty (t^2 + 2\, t)^{-1/2} \, e^{-x\, t} \, dt

Using Watson's Lemma, show the following asymptotic expansion of \fs2 K_0(x)  for \fs2 x\to +\infty:

\fs2K_0(x) \sim e^{-x} \, \sum_{n=0}^\infty (-1)^n\,{\left[\Gamma(n+{1/2})\right]^2 \over2^{n+1/2} \, n!\, \Gamma(1/2)\, x^{n+1/2}}.