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.
2026-04-05 17:10:03.1775409003
is the converse true: in a simply connected domain every harmonic function has its conjugate
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No, to the question as stated by OP in the comments. Take the real part of a holomorphic function on an annulus.
On the other hand, look at the comment by Jonas.