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We all know that Watson's Lemma is used to approximate the integral $$ F\left( s \right)=\int_0^\infty {{e^{ - st}}f\left( t \right)dt} $$ for large $s$. However, for arbitrary $s$, are there any methods to approximate $F(s)$?

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Unsurprisingly, much stronger hypotheses on $f$ would be necessary for this integral to be (absolutely?) convergent for $\Re(s)<0$. Addressing a putative meromorphic continuation, the obvious idea is to integrate by parts, which entails other hypotheses on $f$. I've thought about this kind of thing, but I'd need further details to know what direction to take a response... –  paul garrett Dec 27 '12 at 0:41
When you say approximate, are you interested in something abstractly mathematical or are you trying to compute this numerically? –  Alex R. Dec 27 '12 at 0:46
I prefer abstractly mathematical result. –  widapol Dec 27 '12 at 0:51
Actually, I tried to calculate the following integral $$ {I} = \int_0^\infty {{e^{ - {\mu}x}}g\left( x \right)dx} $$ where $$ g\left( x \right) = \ln \left( {\sum\limits_{i = 1}^n {{\alpha _i}{e^{ - {\mu _i}x}}} } \right) $$ with $\mu,\mu_i, \alpha_i$ are possitive real numbers. Therefore, I though about using Watson's Lemma. However, $\mu$ is abritrarily positive. So the problem became complicated for me. –  widapol Dec 27 '12 at 0:57
If you are trying to approximate the integral for "$s$" essentially fixed, asymptotic expansions cannot give arbitrary precision in general. The proof of Watson's lemma gives a natural estimate for how good an approximation you do get with a particular function. There will be a "turn-around" point where the terms in the asymptotic expansion start getting larger rather than smaller, for fixed $s$, and that's the limit of precision. –  paul garrett Dec 27 '12 at 17:04
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