The question is. Is the converse true: In a simply connected domain every harmonic function has its conjugate? I am not able to get an example to disprove the statement.
Note: as discussed below, there is no mistake here. The notation and conventions have been cleared up for me. However what I have written is not incorrect, either. At least to me, it is just a ...
This is in Section 2.8 of Gilbarg and Trudinger. I believe there are some inaccuracies in the proof supplied, and in any case I think there is a more straightforward proof. Definition: A ...