# I Complex analysis proof

1. May 26, 2017

### Silviu

Hello! I have this Proposition: "A harmonic function is infinitely differentiable". The book gives a proof that uses this theorem: "Suppose u is harmonic on a simply-connected region G. Then there exists a harmonic function v in G such that $f = u + iv$ is holomorphic in G. ". In the proof they present in the book they begin with: "Suppose u is harmonic in G and $z_0 ∈ G$. Let $r > 0$ such that the disk $D[z_0, r]$ is contained in G. " and as a disk is simply connected the conclusion follows from the theorem. My question is, how can you make sure that for any $z_0 \in G$ you can have a disk around $z_0$? (For example if $G=\mathbb{C}$\$\mathbb{R}_{<0}$ and $z_0=0$, you can't find such a disk. What am I missing?

2. May 26, 2017

### FactChecker

Typically, these things are only considered in the interior of the set G. Statements about points on the boundary of G would require a lot of conditions and restrictions.

3. May 26, 2017

### zwierz

they evidently assume $G$ to be an open set:

4. May 26, 2017

### martinbn

Region usually means open and connected.

5. May 26, 2017

### zwierz

by the way, what is a harmonic function at such a point $z_0$?