CHAPTER 3 Complex Differentiation
1. For each following function, determine the singular points at which the function is not analytic. Determine the derivative at the points where the function is analytic using Cauchy-Riemann Equations.
(a) f(z) = 1 / (z + 1)
(b) f(z) = z^2 + z + 1
(c) f(z) = |z|^2
2. Verify that the Cauchy-Riemann equations hold for the following functions and find their derivatives:
(a) f(z) = z^3
(b) f(z) = e^z
(c) f(z) = sin z
3. Suppose u(x, y) = x^2 - y^2.
(a) Show that u is harmonic.
(b) Find a harmonic conjugate function v(x, y).
(c) Express f(z) in terms of z.
4. Show that f(z) = z* (the complex conjugate of z) is not differentiable anywhere.
5. Given f(z) = u + iv is analytic, show that both u and v satisfy Laplace's equation.