I am computing the Hilbert transform of a $2\pi$-periodic function on the interval $[-\pi, \pi]$ of the form $f(x) = |x|^a$, where $a \in \mathbb{R}$.
By definition, we know that the Hilbert transform of this function $f$ is given by $$Hf(x) = \frac{1}{2\pi} \text{p.v.} \int_{-\pi} ^{\pi} f(t) \cot\left({\frac{x-t}{2}}\right)dt$$
Inserting the function and performing some simplifications one sees that
$$Hf(x) = -\frac{1}{2\pi} \text{p.v.} \int_{-\pi} ^{0} t^a \cot\left({\frac{x-t}{2}}\right)dt + \frac{1}{2\pi} \text{p.v.} \int_{0} ^{\pi} t^a \cot\left({\frac{x-t}{2}}\right)dt$$
and so the problem reduces to solving both of these integrals.
I am wondering if there is any way of doing this explicitly? I can't seem to figure it out, and I am wondering if there is a general rule for Hilbert transforms allowing this, or a general way of computing the integral. If not, is there a way to understand the behaviour and asymptotics of these integrals? Is there a better way of doing this using Fourier transforms?
I would appreciate any help.
