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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!

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I don't even think this is true as stated. I feel as though we may need some connectivity conditions on $U$. If we assume that $U$ is simply connected this is not bad, for example. –  Alex Youcis Mar 9 '13 at 23:29
    
@AlexYoucis, thank you for the correction! $U$ must be open, nonempty and connected. I edited my first post. –  Lullaby Mar 9 '13 at 23:32
    
I have posted an answer! –  Alex Youcis Mar 9 '13 at 23:34
    
Just connected is not enough. You need that $U$ is simply connected, otherwise $F$ might not exist. –  mrf Mar 9 '13 at 23:36
    
@mrf There is no issue if you are given the $f$. But yes, we clearly need simply connected if we want to guarnatee the existence of such an $f$. –  Alex Youcis Mar 12 '13 at 16:24

2 Answers 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.)

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Hello mrf, I have the same problem as the OP, but I don't understand your answer, specially when you say "using that $U$ is connected, $a = u + c$. How could you use this assumption to conclude $a= u+c$. Sorry if my question sounds stupid, but I'm not used to work with connectedness of sets. –  Ale Mar 8 at 10:31
    
I've also another question, why can you say that f has a primitive because $U$ is connected, does it follow from some definition? –  Ale Mar 8 at 10:35

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.

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