Prove. Let u(x, y) be harmonic in some region R. Then there exists a function v(x, y) such that f(z) = u(x, y) + iv(x, y) is analytic in R.

Proof. By the definition of a harmonic function u(x, y) satisfies Laplace’s equation. Thus

for all (x, y) in R. Rewriting 1) we get

Now if for two functions M(x, y) and N(x, y) we know that

then M dx + N dy = 0 is an exact differential equation and there exists some function Φ such that dΦ = M dx + N dy. If in 2) we let M = -∂u/∂y and N = ∂u/∂x we see that 2) is an exact differential equation and there exists some function v(x, y) such that

Taking the partial derivative of 3) with respect to x we get

Taking the partial derivative of 4) with respect to y we get

Thus from 4) and 5) we see that u and v satisfy the Cauchy-Riemann equations. Since u(x, y) satisfies Laplace’s equation, the second partial derivatives exist and, consequently, the first partial derivatives ∂u/∂x and ∂u/∂y are continuous in R. Thus the function f(z) = u(x, y) + iv(x, y) is analytic in R.