1. Determine the largest domain Ω on which the function f(z) = log z / z is analytic, where log z is the principal branch of the logarithm. Furthermore, compute f'(z) on your domain Ω.
If u(x, y) = (1/2) ln(x2 + y2) and v(x, y) = tan-1 y/x, then is f(x + iy) = u(x, y) + iv(x, y) analytic on Ω, where Ω is the complex plane with the point z = 0 removed?