(a) Given the integral equation (valid for all real x0)
1 / (1 + x0^2) = (1 / pi) * Integral from -infinity to infinity of u(x) / (x - x0) dx,
use Hilbert transforms to determine u(x0).
(b) Verify that the u(x0) found as your answer to part (a) actually satisfies the integral equation.
(c) From f(z)|y=0 = u(x) + iv(x), replace x by z and determine f(z). Verify that the conditions for the Hilbert transforms are satisfied.
(d) Are the crossing conditions satisfied?
ANS. (a) u(x0) = x0 / (1 + x0^2), (c) f(z) = (z + i)^-1.