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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.

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    yeah, sure. The complex absolute value is nowhere complex differentiable. This makes it pretty much impossible. This was the only hope I had coming a some closed form computation somewhat close. I cannot just say the integral exists. I have to provide a way to get a real number, hence "really".2011-09-05

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