Undergrad Proof of Harmonic Function Infinitely Differentiable

[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?

 

[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.

 

[zwierz]

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

Let r>0r > 0 such that the disk D[z0,r]D[z_0, r] is contained in G.

 

[martinbn]

Region usually means open and connected.

 

[zwierz]

For example if G=CG=\mathbb{C}\R<0\mathbb{R}_{z0=0z_0=0, you can't find such a disk.

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