Existence of harmonic conjugates on simply connected domains via de Rham cohomology

Let $\mathbb{D}\subseteq \mathbb{C}$$\mathbb{D}\subseteq\mathbb C$ be a simply connected domain. Let $u:\mathbb{D}\to \mathbb{R}$$u:\mathbb D\to\mathbb R$ be a harmonic function, then there exists another $v:\mathbb{D}\to \mathbb{R}$$v:\mathbb D\to\mathbb R$ such that $u+iv$$u+iv$ is a holomorphic function.

Proof.

Consider $\ast du=\ast (u_{x}dx+u_{y}dy)=-u_{y}dx+u_{x}dy$$*d u=*(u_xdx+u_ydy)=-u_ydx+u_x dy$.

This is a closed form since $d(-u_{y}dx+u_{x}dy)=-u_{yy}dy\wedge dx+u_{xx}dx\wedge dy=\mathrm{\Delta }udx\wedge dy$$d(-u_ydx+u_x dy)=-u_{yy} dy\wedge dx+u_{xx} dx\wedge dy=\Delta u\, dx\wedge dy$. But since $u$$u$ is harmonic, $d\ast du=0$$d*du=0$.

Thus $\ast du$$*du$ is closed. Since ${\pi }_1(\mathbb{D})=0$$\pi_1(\mathbb D)=0$, we have $H_{\mathrm{d}}^1(\mathbb{D})=0$$H^1_{\mathrm{d}}(\mathbb D)=0$. Hence $\ast du$$*du$ is exact, so there exists a $v$$v$ such that $dv=\ast du$$dv=*du$, i.e.

$$v_{x}dx+v_{y}dy=-u_{y}dx+u_{x}dy⟹v_x=-u_y,v_y=u_x.$$

That is the Cauchy-Riemann condition. $◻$$\square$