Everything proved about entire functions in the previous chapter holds true within a disc for functions analytic in that disc.
There is no generalization for the above results for arbitrary open domains.
Theorem: If $f$ is analytic at $z$, then $f$ is infinitely diffferentiable at $z$.
Proof: Since $f$ is analytic at $z$, it is analytic in an open set containing $z$. So there is some disc containing $z$ where $f$ is analytic, that is, it can be expressed as a power series. And power series’ are infinitely differentiable. $\blacksquare$
Theorem (Uniqueness): Suppose that $f$ is analytic in a region $D$ and that $f(z_n) = 0$ where $(z_n)$ is a sequence of distinct points and $z_n \to z_0 \in D$. Then $f \equiv 0$ in $D$.
Proof: Let $A$ be the set of
Corollary: If two functions $f$ and $g$, analytic in a region $D$, agree at a set of points with an accumulation point in $D$, then $f \equiv g$ through $D$.
Proof: Apply the above theorem to $f - g$. $\blacksquare$
Example: A non-trivial analytic function may have infinitely many zeroes. For example, $\sin z$ is entire an has infinitely many zeroes,