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

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

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

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 $D$.

Proof. The proof is the identity theorem applied to two continuations of the same germ, and its hinge is that simple connectivity makes homotopic curves agree.

Setup. A continuation along a curve $\gamma : [0,1] \to D$ from $z_0$ is a chain of discs and function elements: discs $U_0, \dots, U_m$ covering $\gamma$ with $z_0 \in U_0$, $\gamma(1) \in U_m$, consecutive discs overlapping, and analytic functions $f_0 = f, f_1, \dots, f_m$ with $f_{k+1} = f_k$ on $U_k \cap U_{k+1}$. The endpoint value $f_m(\gamma(1))$ is what “continuation produces” at the far end; and the theorem must prove it independent of the chain chosen.

Step 1: nearby chains agree. If two chains from $z_0$ to $\gamma(1)$ are close; same discs except near the end; their endpoint values agree. The functions agree on the overlap of the two final discs, which contains $\gamma(1)$, an open set with a limit point; by the identity theorem (13.1) the two final functions agree on the whole of their common domain, so their values at $\gamma(1)$ coincide. This is the step that makes “the” continued value meaningful locally.

Step 2: homotopic curves agree, by induction on the number of discs in the homotopy’s uniform cover. Let $\gamma_0, \gamma_1$ be homotopic curves in $D$ with fixed endpoints, and let $H(s,t)$ be the homotopy. Compactness of the image of $H$ gives a finite cover by discs on which continuations exist, and uniform continuity partitions the $(s,t)$ square finely enough that adjacent horizontal paths $\gamma_{s_j}$ are close; within one covering disc at every time. By Step 1, continuation along $\gamma_{s_j}$ and along $\gamma_{s_{j+1}}$ produce the same endpoint value for each $j$; chaining the equalities, the two extreme curves produce the same value.

Step 3: simple connectivity closes the argument. If $D$ is simply connected, every closed curve $\gamma$ in $D$ is homotopic to the constant curve at $z_0$; this is the definition, and it is the only place the hypothesis enters. Continuing $f$ around $\gamma$ therefore produces the same endpoint value as continuing along the constant curve, which is $f(z_0)$: the monodromy; the change in value around a loop; is zero for every loop.

Assembly. Define $F(z)$ for $z \in D$ as the value at $z$ of a continuation of $f$ along any curve from $z_0$ to $z$. Step 3 says the value is independent of the curve; Step 1 says the choice of chain does not matter. $F$ agrees with $f$ near $z_0$, and is analytic; near any $z$, the final disc of any continuation provides a neighbourhood where $F$ equals an analytic function. So $F$ is the single-valued analytic extension. $\blacksquare$

The theorem’s failure on $\mathbb{C}\setminus\{0\}$ is exactly Step 3 failing: a loop around the origin is not homotopic to a point, so the chain of equalities in Step 2 never closes, and $\sqrt z$‘s continuation around the loop returns $-\sqrt z$. Simple connectivity is not a technical convenience; it is the hypothesis that makes Step 2’s chaining terminate in a consistent value.

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.

Mistake 1: Assuming that analytic continuation is always possible. Analytic continuation is not always possible. If a function has a natural boundary (a curve beyond which it cannot be analytically continued), then continuation past that boundary is impossible. For example, the function $\sum_{n=0}^\infty z^{2^n}$ has the unit circle as a natural boundary.

Mistake 2: Forgetting that analytic continuation is unique. If an analytic continuation exists, it is unique. Do not assume that there are multiple distinct analytic continuations of a function to the same domain. The identity theorem guarantees uniqueness.

Mistake 3: Confusing analytic continuation with continuous extension. Analytic continuation requires that the extended function be analytic, not just continuous. A function can be continuously extended to a larger domain without being analytically continuable. For example, $f(z) = |z|^2$ is continuous on $\mathbb{C}$ but not analytic anywhere except at $z = 0$.

Mistake 4: Assuming that the identity theorem holds for non-analytic functions. The identity theorem requires that the functions be analytic. If $f$ and $g$ are merely continuous and agree on a set with a limit point, they need not agree everywhere. For example, $f(x) = 0$ and $g(x) = e^{-1/x^2}$ (with $g(0) = 0$) agree on $\mathbb{R}$ but are not equal as functions on $\mathbb{C}$.

Mistake 5: Forgetting that analytic continuation may introduce singularities. When analytically continuing a function to a larger domain, new singularities may appear. For example, the Gamma function is analytic for $\mathrm{Re}(z) > 0$, but its analytic continuation to $\mathbb{C}$ has poles at $z = 0, -1, -2, \dots$. Always check for singularities in the extended domain.

• Taylor and Laurent Series: Power series provide local representations that can be analytically continued to larger domains.

• Liouville’s Theorem: Liouville’s theorem characterizes entire functions and is used in proving uniqueness of analytic continuation.

• Complex Functions and Analyticity: Analyticity is the essential property that enables continuation beyond the original domain.

• Classical Mechanics

• Electromagnetism

• Statistical Learning

• Statistical Mechanics

• Applications of Contour Integration
• Argument Principle and Rouché’s Theorem
• Cauchy’s Integral Formula
• Cauchy’s Theorem