The equation for and electric field is given by $E=−∇ψ$ where $\psi$ is the potential, and in this case $ψ=−Q\ln r$ where $Q$ is just some constant. I have found its harmonic conjugate to be $−Qθ+c$ where $c$ is some constant. What does it say about the field? I know that if I calculate the field directly from $E=−∇ψ$, I get $E=Q/r$ pointing radially outwards, but I am not sure how to interpret the harmonic conjugate found (is it right?).
Please help! Thank you.
Added: I can see that $\psi$ is only dependent on $r$ and its h.c. only dependent on $\theta$ means that they are orthogonal, but what else about $E$ can I tell from the h.c. of $\psi$? Perhaps I have made a mistake but should the h.c. have a factor $1\over r$ in the term with the $\theta$?
Update: Thank you to everyone who read this question. I believe I have figured it out now. The h.c. I have obtained is correct. The correct interpretation follows directly from the CR equations! I don't know why I didn't see this earlier...