Analytic Continuation

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 $f_1$.

Uniqueness. If an analytic continuation exists, it is unique. This follows from the identity theorem: any two analytic continuations of $f_1$ to the same domain must agree everywhere on that domain.

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, $E = D$. $\blacksquare$

Consequence. If two analytic functions agree on any interval of $\mathbb{R}$, on any open subset, or on any set with an accumulation point, then they are identical wherever both are defined. This is the core reason why analytic continuation is unique.

The Gamma function $\Gamma(z)$ provides a classic example. The integral representation

$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,dt$

converges only for $\mathrm{Re}(z) > 0$. However, using the functional equation $\Gamma(z+1) = z\Gamma(z)$, we can extend $\Gamma(z)$ meromorphically to the entire complex plane with simple poles at $z = 0, -1, -2, \dots$.

Continuation via the functional equation. For $\mathrm{Re}(z) > -1$, $z \neq 0$, define $\Gamma(z) = \Gamma(z+1)/z$. Repeating this pushes the domain leftward strip by strip, producing a meromorphic function on $\mathbb{C}$.

Reflection formula. The analytic continuation satisfies Euler’s reflection formula: $\Gamma(z)\Gamma(1-z) = \pi / \sin(\pi z)$, which relates values across the whole plane.

The Riemann zeta function is defined for $\mathrm{Re}(s) > 1$ by the Dirichlet series

$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$

This series diverges for $\mathrm{Re}(s) \leq 1$. However, $\zeta(s)$ admits an analytic continuation to $\mathbb{C} \setminus \{1\}$ with a simple pole at $s = 1$.

Continuation via the functional equation. The Riemann zeta function satisfies the functional equation

$\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)$

which relates $\zeta(s)$ to $\zeta(1-s)$ and provides the continuation to $\mathrm{Re}(s) < 0$. The trivial zeros at $s = -2, -4, -6, \dots$ arise from the sine factor.

Connection to prime numbers. The Euler product $\zeta(s) = \prod_p (1 - p^{-s})^{-1}$ for $\mathrm{Re}(s) > 1$ links the zeta function to the distribution of primes, culminating in the Riemann hypothesis about the non-trivial zeros.

Not every function defined on a domain can be analytically continued to a larger domain. A natural boundary is a curve beyond which analytic continuation is impossible.

Example. The function $f(z) = \sum_{n=0}^\infty z^{2^n}$ has the unit circle $|z| = 1$ as a natural boundary. The series converges for $|z| < 1$, but every point on $|z| = 1$ is a singularity (the function is unbounded near a dense set of points).

Lacunary series. More generally, a power series $\sum a_k z^{n_k}$ with gaps satisfying $n_{k+1}/n_k > 1 + \delta$ has the circle of convergence as a natural boundary (Hadamard gap theorem).

Theorem 13.2 (Monodromy). Let $f$ be analytic on a simply connected domain $D$ and suppose $f$ can be analytically continued along every curve in $D$. Then $f$ extends to a single-valued analytic function on all of $D$.

Intuition. The monodromy theorem guarantees that local analytic continuation along paths gives a well-defined global function if the domain is simply connected. On multiply connected domains, continuation around a closed loop may produce a different branch (monodromy).

Branch points. The function $\sqrt{z}$ has a branch point at $z = 0$. Continuing along a loop encircling the origin changes the sign of the function. The monodromy theorem fails because $\mathbb{C}\setminus\{0\}$ is not simply connected.

Theorem 13.3 (Schwarz Reflection). Let $f$ be analytic on a domain $D$ symmetric about the real axis, with $f$ real-valued on $D \cap \mathbb{R}$. Then $f(\overline{z}) = \overline{f(z)}$ for all $z \in D$.

Application to continuation. If $f$ is analytic on the upper half-plane and real on the real axis, the Schwarz reflection principle extends $f$ to the lower half-plane. This provides a method of continuation for many special functions.

Show that $f(z) = \sum_{n=0}^\infty z^n$ defined on $|z| < 1$ has an analytic continuation to $\mathbb{C} \setminus \{1\}$.

Solution. On $|z| < 1$, $f(z) = 1/(1-z)$. The function $F(z) = 1/(1-z)$ is analytic on $\mathbb{C} \setminus \{1\}$ and agrees with $f$ on $|z| < 1$. Therefore $F$ is the analytic continuation of $f$. The function $f$ can only be defined on $|z| < 1$ by its power series, but the continued function exists almost everywhere in the complex plane.

Find the analytic continuation of $f(z) = \int_0^\infty e^{-t} t^{z-1}\,dt$ from $\mathrm{Re}(z) > 0$ to $\mathrm{Re}(z) > -1$.

Solution. The integral converges for $\mathrm{Re}(z) > 0$. Using the identity $\Gamma(z) = \Gamma(z+1)/z$, we extend the definition: for $\mathrm{Re}(z) > -1$, $z \neq 0$, define $f(z) = \int_0^\infty e^{-t} t^{z}\,dt / z$. This agrees with the original definition where both are valid and extends the domain left by one strip.

1. Prove that $f(z) = \sum_{n=0}^\infty z^{n!}$ has the unit circle as a natural boundary.
2. Find the analytic continuation of the geometric series $\sum_{n=0}^\infty (-1)^n z^{2n}$ defined on $|z| < 1$.
3. Show that the function $\sum_{n=1}^\infty \frac{z^n}{n^2}$ has no singularity on $|z| = 1$ and can be continued to a larger domain. Where are its singularities?
4. Determine all possible analytic continuations of $\sqrt{z}$ to the complex plane. Explain the role of branch cuts.
5. Use the Schwarz reflection principle to show that if $f$ is analytic on $\mathbb{C}\setminus \mathbb{R}$ and continuous on $\mathbb{R}$, then $f$ is entire.

Analytic continuation extends analytic functions to larger domains uniquely. The identity theorem guarantees uniqueness. The Gamma and zeta functions are classical examples. Natural boundaries prevent continuation past essential singularities. The monodromy theorem governs when continuation around loops is single-valued. The Schwarz reflection principle extends functions across the real axis. These ideas are essential for understanding special functions and their properties throughout complex analysis.