4

I recently encountered the following technique in a book on differential equations.

Suppose we know the asymptotic expansion for the Laplace transform $F(q)$ of some function $f(z)$ for large $q$. To get an asymptotic expansion of $f(z)$ for small $z$ we simply apply the inverse Laplace transform to the asymptotic series of $F(q)$ term by term.

This was done in an informal way in the text, and I'm not that familiar with the theory of asymptotic expansions. Can someone tell me how legitimate this procedure is? Specifically, provided the term by term inversion makes sense, how certain can I be, in general, that I end up with an asymptotic expansion of $f(z)$? Appropriate references also welcome in lieu of answers.

Billy Pilgrim
  • 509
  • 2
    The method is essentially the converse of Watson's lemma. See this question for details on when it is valid. While the question in the link concerns only the first term in the expansion, the result can be extended to higher order terms under appropriate conditions. – Antonio Vargas Aug 29 '16 at 11:28

0 Answers0