Analytic Continuation

13.1 Definition

Definition. If $f_1$ is analytic on $D_1$ and $f_2$ is analytic on $D_2$ with $D_1 \cap D_2 \neq \emptyset$ and $f_1 = f_2$ on $D_1 \cap D_2$Then $f_2$ is an analytic Continuation of $f_1$.

13.2 Identity Theorem

Theorem 13.1 (Identity Theorem). If $f$ and $g$ are analytic on a domain $D$ and agree on a set With a limit point in $D$Then $f = g$ on all of $D$.

Proof. Let $E = \{z \in D : f^{(n)}(z) = g^{(n)}(z) \mathrm{\ for\ all\ } n \geq 0\}$. $E$ is Non-empty (it contains the limit point by continuity of derivatives). $E$ is closed (by continuity). If $z_0 \in E$The Taylor series of $f$ and $g$ at $z_0$ coincide, so $f = g$ in a neighbourhood of $z_0$Giving $E$ open. Since $D$ is connected, $E = D$. $\blacksquare$