I'm working from Bender and Orszag's "Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory" and I am trying to solve problem 6.48(b):
Find the leading order behaviour of
$\left(\frac{d^n}{dx^n}\right) \left.\Gamma(x)\right|_{x=1}$ as $n \rightarrow \infty$.
The section the question corresponds to is on Laplace type integrals and Watson's lemma, but I haven't been able to pull off anything using that material.
Any suggestions would be greatly appreciated!
This isn't a complete answer, but perhaps an outline will help you.
One approach would be to follow the steps
$$ \Gamma^{(n)}(1) \sim \int_0^1 (\log t)^n e^{-t}\,dt \sim \int_0^1 (\log t)^n\,dt = (-1)^n \Gamma(n+1). $$
Neither $\sim$ is really a direct application of Laplace/Watson, but the ideas behind them are the same.