All Complex Analysis Resources
Example Question #1 : Analytic And Harmonic Functions
Find a Harmonic Conjugate of
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?
The Entire Complex Plane
Rewriting in real and complex components, we have that
So this implies that
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.