10.3. Laplace equation and conjugate harmonic functions

a. Show that the real and imaginary parts of an analytic function of a complex variable z f (z) = (x, y)+i(x, y)
are both harmonic functions, i.e. they satisfy the Laplace equation ∇2 = ∇2 = 0.

b. Vice-versa, use the Cauchy–Riemann equations to show that if (x, y) is a function that satisfies the Laplace equation, then there exists another harmonic function (x, y) (called the conjugate function of ) such that f (z) = +i is an analytic function of complex variable.