I am reading Hilbert transform recently and meet two questions. The book I am reading is Debnath and Bhatta "Integral Transforms and Their Applications".
If we define the Hilbert transform on the real line is, for $x \in \mathbb{R}$, $$H_x\{f(t)\}=\frac{1}{\pi}PV\int_{-\infty}^\infty \frac{f(t)}{t-x}dt=\lim_{\epsilon \rightarrow 0}\left(\int_{-\infty}^{t-\epsilon} +\int_{t+\epsilon}^\infty\right)\frac{f(t)}{t-x}dt,$$ where PV stands for the Cauchy principal value. Then the Fourier transform of the Hilbert transform, which can be considered as a convolution, is $$F_k\{H_x\{f(t)\}\}=i sgn(k) F_k\{f(t)\},$$ where $i^2=-1$, $sgn(x)=\begin{cases} 1, x>0,\\ -1,x<0 \end{cases}$ and $F_k\{f(t)\}=\frac{1}{\sqrt{2 \pi}}\int_0^\infty e^{-ikt}f(t)dt $, since $F_k\{\sqrt{\frac{2}{\pi}}\left(-\frac{1}{x}\right)\}=i sgn(k)$.
The first question, Eq.(9.4.3), if we let a complex variable $z=x+i y$, then why $$F_k\{H_z\{f(t)\}\}=2 i e^{-ky} H(k) F_k\{f(t)\},$$ where $H(k)=\frac{1}{2}(1+sgn(k))$. Which variable are we transforming here, $x\rightarrow k$ or $z\rightarrow k$?
The second question appears in the Mellin transform of the Hilbert transform. There is an integral I do not know how to solve it: $$PV\int_0^\infty\frac{x^{p-1}}{t-x}dx=\pi Cot(\pi p).$$ I understand we need to separate the integral or use the residue theorem to get rid of the singularity. But I cannot go further.
Thanks in advance.