Analytic and Harmonic Functions - Complex Analysis
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Question
Find a Harmonic Conjugate 
of 
Find a Harmonic Conjugate
Answer
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.
where
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Question
Given 
, where does 
exist?
Given
Answer
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.
Rewriting
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
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