I currently face a less appealing integral which emerged computing the expectation of some random variable. It reads as (omitting all unnecessary constants except $\alpha\in(0,1)$)
$ PV \int_{-\infty}^\infty \frac{\exp[1-\exp[iu-|u|^\alpha(1-i\text{sgn}(u)]]}{u}du.$
It converges as a principal value integral. However, I really have to compute it. Numerically this is not a trivial task, because it does not satisfy the Hoelder condition around zero.
First thought: Contour integration. Could I replace $\text{sgn}(u)$ with $\frac{u}{|u|}\mathbb{1}_{u\neq 0}$? Due to the $\text{sgn}$-function we don't have a simple pole: $z\frac{\exp[1-\exp[iz-|z|^\alpha(1-i\frac{z}{|z|}\mathbb{1}_{z\neq 0}]]}{z}$ is not holomorphic in zero. Computing the residue will be a pain, I guess. What is the best way to attack this problem?
Idea: Compute $a_{-1} = \int_{\gamma}f(z)dz$, where $\gamma$ is a suitable contour and use series expansion of the exponential function. Or is it possible to obtain the Laurent series?
Thank you for your comments.