Asymptotic expansion question

Question

How may I use Watson's Lemma to find the full asymptotic expansion for;

$$ I(\lambda)= \int_0^\infty e^{-\lambda(1+s)}ln(1+s^2)ds $$

as $\lambda \rightarrow \infty$.

Thanks in advance

Answer 1

Consider the function $$f(t) = \int_0^\infty e^{-s} \ln(1 +t^2s^2) ds,$$ We have $f(0) =0, f'(0)=0$ and $$f''(0) = 2\int_0^\infty e^{-s} s ds =2$$ hence $f(t) = t^2 + o(t^2)$ when $t\to 0$

By changing variable, $$I(\lambda) = e^{-\lambda} \lambda^{-1} f(\lambda^{-1}) = \frac1{e^\lambda \lambda^3} + o\left(\frac1{e^\lambda \lambda^3}\right).$$