I am looking for help calculating
$$ I(k,y) = \int_1^\infty \frac{\textrm{e}^{-\textrm{i} k x}}{\sqrt{(x-\textrm{i} y)^2 -1}} \textrm{d} x, $$
where we assume that $k$ has small, negative imaginary part and $y$ is real and positive.
There is an expression in the case $y=0$ from the Digital Library of Mathematical Functions: (picture here) but I can't relate this to $I(k,y)$ without introducing an extra integral to calculate.
I've tried contour integration, expanding the square root function as a Taylor series, and comparing with other Bessel, Hankel and Struve identities, but nothing seems to work.