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.
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.
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.
The answer is yes. If a domain is not simply connected, you can always construct a harmonic function without a harmonic conjugate there. For example, for an annulus centred at the origin, take $f(x,y)=\log\sqrt{x^2+y^2}$ in . This cannot have a harmonic conjugate, as if it did you would get a branch of the logarithm analytic in such a domain.
You can look at a nice explanation for the construction in the general case here:
Show $\Omega$ is simply connected if every harmonic function has a conjugate