Analytic and Harmonic Functions
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Complex Analysis › Analytic and Harmonic Functions
Questions 1 - 2
1
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.
2
Given , where does
exist?
Nowhere
The Entire Complex Plane
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.