Solution
= Solution
<Watson lemma> states that if, near zero,
$$
f(t)\sim\sum_{n=0}^\infty a_nt^{n+\lambda-1},
\qquad \lambda>0,
$$
and the tail is suitably controlled, then
$$
\int_0^b e^{-xt}f(t)dt
\sim\sum_{n=0}^\infty a_n\Gamma(n+\lambda)x^{-n-\lambda}
$$
as $x\to+\infty$.