Assume , where has an infinite number of derivatives in the neighborhood of , with , and .
Suppose, in addition, that
Then, it is true that
is finite , and the following asymptotic equivalence holds:
Proof: See, for instance, Watson (1918) for the original proof or Miller (2006) for a more recent development.
Example
Find a simple asymptotic approximation for
for large values of , that is when .
Solution: By direct application of Watson's lemma, with and ,
so that for . It is easy to see that:
This development used the fact that , a known property of the Gamma function.
References
Miller, P.D. (2006), Applied Asymptotic Analysis, Providence, RI: American Mathematical Society, p. 467, ISBN9780821840788.
Watson, G. N. (1918), "The harmonic functions associated with the parabolic cylinder", Procedings of the London Mathematical Society, vol. 2, no. 17, pp. 116–148.