Complex Analysis : Analytic and Harmonic Functions

Example Questions

Example Question #1 : Analytic And Harmonic Functions

Find a Harmonic Conjugate  of

Explanation:

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.

Example Question #2 : Analytic And Harmonic Functions

Given , where does  exist?

Nowhere

The Entire Complex Plane

Nowhere

Explanation:

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.