Identity Theorem and Analytic Continuation

The identity theorem expresses the remarkable rigidity of holomorphic functions: they are completely determined by their values on any set with a limit point.

The Identity Theorem

Theorem4.11Identity theorem

Let $f$ and $g$ be holomorphic on a connected domain $D$ . If $f(z_n) = g(z_n)$ for a sequence $\{z_n\} \subset D$ with an accumulation point $z^* \in D$ , then $f \equiv g$ on $D$ .

Equivalently, if $f$ is holomorphic and not identically zero on a connected domain, then the zeros of $f$ are isolated.

Proof

Set $h = f - g$ . By continuity, $h(z^*) = 0$ . Write $h(z) = \sum_{n=0}^\infty a_n(z-z^*)^n$ near $z^*$ .

If not all $a_n$ are zero, let $m$ be the smallest index with $a_m \neq 0$ . Then $h(z) = (z-z^*)^m \phi(z)$ with $\phi(z^*) = a_m \neq 0$ . By continuity of $\phi$ , there exists $r > 0$ such that $\phi(z) \neq 0$ for $|z-z^*| < r$ . But $h(z_n) = 0$ with $z_n \to z^*$ , so $(z_n - z^*)^m \phi(z_n) = 0$ with $z_n \neq z^*$ eventually, giving $\phi(z_n) = 0$ — a contradiction.

Therefore all $a_n = 0$ , so $h \equiv 0$ near $z^*$ . The set $S = \{z \in D : h^{(n)}(z) = 0 \text{ for all } n\}$ is closed (by continuity) and open (by the power series argument above). Since $D$ is connected and $S \neq \emptyset$ (it contains $z^*$ ), we have $S = D$ . $\blacksquare$

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Analytic Continuation

Definition4.5Analytic continuation

Let $f_1$ be holomorphic on a domain $D_1$ and $f_2$ holomorphic on $D_2$ with $D_1 \cap D_2 \neq \emptyset$ and connected. If $f_1 = f_2$ on $D_1 \cap D_2$ , then $f_2$ is called an analytic continuation of $f_1$ to $D_2$ . By the identity theorem, such a continuation is unique.

ExampleAnalytic continuation of the Gamma function

The Gamma function $\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}\,dt$ converges for $\text{Re}(z) > 0$ . Using the functional equation $\Gamma(z+1) = z\Gamma(z)$ , we define

$\Gamma(z) = \frac{\Gamma(z+1)}{z}$

which extends $\Gamma$ to $\text{Re}(z) > -1$ (except $z = 0$ ). Iterating: $\Gamma(z) = \frac{\Gamma(z+n)}{z(z+1)\cdots(z+n-1)}$ extends $\Gamma$ to $\mathbb{C} \setminus \{0, -1, -2, \ldots\}$ .

Monodromy and Multi-Valuedness

Definition4.6Monodromy theorem

If $f$ can be analytically continued along every path in a simply connected domain $D$ from $z_0$ , then the result is a single-valued holomorphic function on $D$ .

On a non-simply connected domain, analytic continuation along different homotopy classes of paths may yield different values, leading to multi-valued functions and Riemann surfaces.

ExampleMonodromy of the logarithm

The function $\log z$ can be analytically continued along any path in $\mathbb{C} \setminus \{0\}$ . Continuing around the origin: starting with $\log(1) = 0$ , after a full counterclockwise circuit, $\log(1) = 2\pi i$ . Each additional loop adds $2\pi i$ :

$\log z = \ln|z| + i\arg(z) + 2\pi i k, \quad k \in \mathbb{Z}.$

The monodromy group is $\mathbb{Z}$ , and the Riemann surface is a helicoid.

RemarkUniqueness of analytic continuation

The identity theorem guarantees that analytic continuation along a fixed path is unique. The subtlety arises only when different paths lead to different values (monodromy). This is the origin of Riemann surfaces and is connected to the fundamental group of the domain.