Analytic and Harmonic Functions

is said to be a harmonic conjugate of if their are both harmonic in their domain and their first order partial derivatives satisfy the Cauchy-Riemann Equations. Computing the partial derivatives

where is any arbitrary constant.

Rewriting in real and complex components, we have that

So this implies that

where

Therefore, checking the Cauchy-Riemann Equations, we have that

So the Cauchy-Riemann equations are never satisfied on the entire complex plane, so is differentiable nowhere.