Harmonic functions and all-pure functions

It is clear that the real and imaginary parts of pure functions are harmonic functions. Naturally, ask a question:

If the $u$ is a harmonic function on the area $d$, is there necessarily a function $f\in H (D) $ so that $${\RM re}f=u$$ is established?

Generally speaking, this conclusion is wrong. But if the restricted area $d$ is single-connected, then the conclusion is right. The following is a proof of this conclusion: note $\delta u=0$, and $d$ single-connection, thus $$-u_{y}{\rm d}x+u_{x}{\rm d}y$$ Is the full differential of a function, so that the integral is independent of the path, so that the $ $v (x, y) =\int_{(x_{0},y_{0})}^{(x, y)}-u_{y}{\rm d}x+u_{x}{\rm d}y$$

Then consider the function $f=u+iv$, wherein the $u,v$ are actually micro and satisfy the Cauchy-riemann equation, thus $f\in H (D) $.

Note: If $u$ is required to be the imaginary part of a fully pure function, then consider the function $if$.

Harmonic functions and all-pure functions