I'm hoping to evaluate the contour integral
$$\int_\gamma\frac{e^{-\sqrt{z}}}{(z-b)^{k+1}}dz$$
for $b > 0$, $k\in{\bf Z}_+$ and $\gamma$ a closed contour of the reader's choice which contains $b$.
My complex analysis is really rusty, but my understanding is that since $b$ is strictly positive, then we can choose the branch of the square root to be the negative real-axis and then make the contour small enough to avoid crossing this branch, and thus we don't have to worry about the branch cut when evaluating the integral, is that correct?
Also, does this integral have a closed form? Essentially my goal is to use it as a method to quickly evaluate the $k^{th}$ derivative of $e^{-\sqrt{z}}$.
I was thinking of writing the exponential function as a power series and then breaking the integral up and evaluating each of the resulting integrals individually,
$$\sum_{n=0}^\infty\frac{(-1)^n}{n!}\int_\gamma\frac{z^\frac{n}{2}}{(z-b)^{k+1}}dz$$
hoping that by the residue theorem most of these integrals will be zero, but like I said my complex analysis is really rusty, so I was hoping someone could help me do this.