Harmonic function, existence of a constant

Question

May i ask you for a little help about a problem with harmonic function? It seems to be not that difficult, in a way even intiutively obvious but i don't really know how to show this explicitly.

We have $U\subset \mathbb{R}^{2}$ open, non-empty, connected subset and a harmonic function $u: U\rightarrow \mathbb{R}$ given by $f:=\frac{\partial u}{\partial x} - i \frac{\partial u}{\partial y}$. Let $F$ be a holomorphic function $U\rightarrow \mathbb{C}$ with $F'=f$. Show that there exist an $c\in \mathbb{R}$ such that $\operatorname{Re}(F) = u + c$.

Okay, $f$ is given only through $u$. This means that after the integration of $F$ i will have an expression with the real part of $u(x,y)$.

I would be glad if someone could help me to move on. Thank you in advance!

Answer 1

Hint: Apply the Cauchy-Riemann equations to conclude that the total derivative of $\text{Re}(F)$ is equal to the total derivative of $u$. Use then connectivity and the Intermediate Value Theorem to conclude that $u$ and $\text{Re}(F)$ differ by a constant.

Answer 2

Your formulation is a little strange. I assume that you start with a harmonic function $u$ and then define $f$.

First note that $f$ is holomorphic since it satisfies Cauchy-Riemann's equations. Assuming that $U$ is simply connected (and this assumption is essential), $f$ has a primitive, $F$. Again, using Cauchy-Riemann, it follows that if $F=a+ib$, you have that $a'_x = \operatorname{Re} f = u'_x$ and similarly $a'_y = u'_y$, so using that $U$ is connected, $a = u+c$.

(This is one common way to prove the existence of harmonic conjugates.)