Suppose an electric field $E=-\nabla \psi$ where $\psi=-Q\ln r$ where $Q$ is just some constant and I have found its harmonic conjugate to be $-Q\theta+c$ where $c$ is some constant. What does it say about the field? I know that if I calculate the field directly from $E=-\nabla \psi$, I get $E=Q/r$ pointing radially outwards, but I am not sure how to interpret the harmonic conjugate found (is it even right?).
Update: I believe I have figured this out now. All the correct information follows from the Cauchy-Riemann equations!