= Solution
For a <harmonic function> $u$, put $g=u_x-iu_y$. Its <Wirtinger derivative> is
$$
\frac{\partial g}{\partial\overline z}
=\frac12\bigl(u_{xx}+u_{yy}+i(u_{xy}-u_{yx})\bigr)=0,
$$
so the <Cauchy-Riemann equations> make $g$ holomorphic. In the <simply connected domain> it has a single-valued primitive
$$
F(z)=\int_{z_0}^{z}g(\zeta)\,d\zeta.
$$
The integral is <path independent>: local <holomorphic primitives> make its value invariant under <path> <homotopies>, and being <simply connected> makes any two <paths> with the given endpoints <homotopic>, as in Question 3.
Since $F'=g$, the <real part> satisfies $(\operatorname{Re}F)_x=u_x$ and $(\operatorname{Re}F)_y=\operatorname{Re}(ig)=u_y$. Hence $\operatorname{Re}F-u$ is constant on the <connected> domain. Adding a real constant corrects it, producing \b[a single-valued holomorphic $f$ with $\operatorname{Re}f=u$]. The <imaginary part> is a <harmonic conjugate> and is unique up to an additive real constant. This is the <global harmonic conjugate criterion by vanishing periods> in the <simply connected> case.
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