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Can someone tell me if the following proposition is true ?

Theorem If $u=g + i h$ is a holomorphic function in $\Omega\subseteq \mathbb{C}$ and $\Omega$ is simply connected, then $v(z)=u(w)+ \int_\gamma \,g_x(z)-ih_y (z) \,dz$ is a primitive function of $u$ (where $w\in \Omega$ is fixed and $\gamma$ is some path from $w$ to $z$).

(I have come across (the implicit use of) this proposition by reading about something not really related to complex analysis and since I know very little about it, if wondered if it actually would be true taken out of context like this. I also wouldn't mind a proof, if it is true and someone would have the time.)

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up vote 3 down vote accepted

I shall assume that $g$ and $h$ are realvalued. Since $u:=g+ih$ is holomorphic it follows from the the CR equations that $h_y=g_x$. Therefore for any curve $\gamma\subset\Omega$ connecting the point $z_0$ with a variable point $z$ one has $$\int_\gamma (g_x- i h_y)\ dz=(1-i)\int_\gamma g_x\ (dx+i dy) =(1-i)\int_\gamma(g_x\ dx + i h_y dy)=(1-i)\Bigl(g(z)-g(z_0)+i\bigl(h(z)-h(z_0)\bigr)\Bigr)=(1-i)\bigl(u(z)-u(z_0)\bigr)\ .$$ It follows that $$v(z):=u(z_0)+\int_{z_0}^z (g_x- i h_y)\ dz=i u(z_0)+(1-i) u(z)\ ,$$ which shows that your $v$ is more or less the given $u$ again, and not a primitive of $u$.

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