Let f be the function f(z) = 1/z and let A and B be the regions A = {z: 2 < |z| < 4} and B = {z: -π/4 < Arg(z+1) < π/4}.
(i) State whether or not the function f is bounded on each of the sets A and B. Justify your answers briefly.
(ii) For each of the sets A and B, state whether or not ∮f(z)dz = 0 for every closed contour C in the region. Justify your answers briefly.
(b) Let f be the function f(z) = z - z + |z|^3.
(i) Prove that f(x+iy) = x + x + y^3/2 + iy.
(ii) Prove that f is not analytic at any point of C.
(iii) Find the derivative of f, specifying its domain.
