Consider the following integral:
$$ \int_{0}^{\infty} e^{-xt} \ln(1+\sqrt{t})dt $$
Calculate its asymptotic expansion to ALL orders as $x\rightarrow\infty$.
It seems the natural thing to do is expand the integrand as a Taylor series and integrate term-by-term. I've been given the hint that I can express difficult integrals in terms of the Gamma function. I also am required to discuss the convergence of the resulting series.
$$ e^{-xt}=\sum_{k=0}^{\infty}\frac{(-xt)^{k}}{k!} $$
$$ \ln(1+\sqrt{t})=\sum_{k=1}^{\infty} (-1)^{k+1} \frac{t^{k/2}}{k} $$
I understand how to use this to get the leading order behavior, but how to get the behavior at all orders?