# Definition clarification of analytic continuation of holomorphic function

I was given the definition of anlytic continuation as in Stein's book- given two regions $$\Omega\subset \Omega'$$ for analytic functions $$F:\Omega'\to \mathbb{C}$$ is an analytic continuation of $$f:\Omega \to \mathbb{C}$$ if $$F$$ agrees with $$f$$ on $$\Omega$$. I know that $$\sin:\mathbb{C}\to \mathbb{C}$$ is an analytic continuation of $$\sin:\mathbb{R}\to \mathbb{R}$$ because they agree on $$\mathbb{R}$$, but this is not a region in $$\mathbb{C}$$, so how does that add up with the defintion?

## 1 Answer

There is some ambiguity here (not in what you wrote, which is perfectly clear, but in Stein's textbook). Sometimes, while writing about analytic continuation, Stein indeed assumes that $$\Omega$$ and $$\Omega'$$ are regions. But when finally Stein states a theorem about analytic continuation, the statement is:

Theorem: Suppose $$f$$ is a holomorphic function in a region $$\Omega$$ that vanishes on a sequence of distinct points with a limit point in $$\Omega$$. Then $$f$$ is identically $$0$$.

As you can see, when it is stated like this, the theorem on analytic continuation applies indeed to the situation that you described.

• I think it answered my question; We know that $\sin(z)$ agrees with $\sin(x)$ on $\mathbb{R}$, and I want to show that this extantion is unique- this is where I use the theorem you mentioned, correct? Thanks for your answer! – Simon Green Jan 18 at 11:10
• Yes, that is correct. – José Carlos Santos Jan 18 at 11:20