6. Hilbert transforms
Given
$$v[x] = -\frac{P}{\pi} \int_{-\infty}^{\infty} \frac{u[s]}{s - x} ds = \frac{1}{1 + x^2}$$
(6.130)
determine $u[x]$ and construct the corresponding analytic function $f[z] = u[x, y] + iv[x, y]$.
Then verify that $f[z]$ satisfies the necessary requirements for $\{u, v\}$ to constitute a Hilbert
transform pair. Note that $u[x] = u[x, 0]$ and $v[x] = v[x, 0]$.
