Analytic Continuation

Q: What is analytic continuation?

Analytic continuation is a technique for extending a holomorphic function defined on some region to a larger region while preserving holomorphy. By the identity theorem, if two holomorphic functions agree on a common open set they agree everywhere, so the extension is unique. The basic method is to chain overlapping power-series disks of convergence.

Q: What is the monodromy theorem?

The monodromy theorem guarantees that analytic continuation within a simply connected domain D does not depend on the path. That is, if two paths in D are homotopic, the results of continuing along them coincide. In non-simply-connected domains path dependence can occur, and multivalued functions arise.

Q: How are multivalued functions related to analytic continuation?

Multivalued functions such as log z and z^{1/2} change value when continued analytically along a path that encircles a branch point (z = 0, ∞), even after returning to the start. log z shifts by 2πi per loop, and z^{1/2} changes sign after one loop. Viewing them on a covering space called a Riemann surface makes them single-valued holomorphic functions. Multivaluedness and analytic continuation are two sides of the same coin.

Q: What does analytic continuation of the zeta function mean?

The Riemann zeta function ζ(s) = Σ n^{-s} is defined by a Dirichlet series that converges for Re(s) > 1, but analytic continuation extends it to all of C except for a simple pole at s = 1. The functional equation ζ(s) = 2^s π^{s-1} sin(πs/2) Γ(1-s) ζ(1-s) is a consequence of this continuation and sets the stage for the Riemann hypothesis.

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Published: 2026-03-20 · Updated: 2026-06-10

1. The Identity Theorem and the Idea of Analytic Continuation

Theorem (Identity Theorem)

Let $D$ be a connected open set and let $f, g$ be holomorphic functions on $D$. If $f = g$ on a set $S$ that has an accumulation point in $D$, then $f = g$ on all of $D$.

The identity theorem says that a holomorphic function is completely determined by its values on a "small piece" of the region. This is the foundation of analytic continuation.

Definition (Analytic Continuation)

Let $D_1 \subset D_2$ be connected open sets and let $f_1$ be holomorphic on $D_1$. If a holomorphic function $f_2$ on $D_2$ satisfies $f_1 = f_2$ on $D_1$, then $f_2$ is called the analytic continuation of $f_1$ to $D_2$.

Example: Geometric Series

The function $f_1(z) = \displaystyle\sum_{n=0}^{\infty} z^n$ converges only for $|z| < 1$, and its sum equals $\dfrac{1}{1-z}$. On the other hand $f_2(z) = \dfrac{1}{1-z}$ is holomorphic on all of $\mathbb{C} \setminus \{1\}$. Since $f_1 = f_2$ on $D_1 = \{|z|<1\}$, $f_2$ is the analytic continuation of $f_1$ to $\mathbb{C} \setminus \{1\}$.

Corollary (Uniqueness of Analytic Continuation)

By the identity theorem, if an analytic continuation of $f_1$ to $D_2$ exists, then it is unique. Indeed, any two continuations $f_2, \tilde{f_2}$ agree on $D_1$, hence agree on all of $D_2$.

What Analytic Continuation Means

A power series diverges outside its disk of convergence, but that only means the series representation breaks down; the function itself may still exist. Analytic continuation provides a way to capture the "true extent" of a function beyond any single series representation. Note, however, that this uniqueness holds for continuation along a fixed path. Different paths can yield different values (monodromy), which is the subject of Sections 3 and 4.

2. Continuation by Chaining Power-Series Disks

The most basic method of analytic continuation is to extend the function by chaining overlapping disks of convergence of power series (Weierstrass's method).

Start from the disk of convergence $D_0$ of a power series $\displaystyle\sum a_n (z - z_0)^n$ centered at $z_0$.
Re-expand the Taylor series at another point $z_1 \in D_0$ to obtain a new disk of convergence $D_1$.
If $D_1$ extends beyond $D_0$, the function is extended to $D_0 \cup D_1$.
Repeat this procedure, advancing the continuation along a path.

Concretely, the re-expansion in the second step is

$$f(z) = \displaystyle\sum_{n=0}^{\infty} b_n (z - z_1)^n, \qquad b_n = \dfrac{f^{(n)}(z_1)}{n!}$$

If the radius of convergence of this new series exceeds $|z_1 - z_0|$, then $f$ has been extended beyond the original disk of convergence.

Figure 1. Analytic continuation via a chain of power-series disks. On the overlap of each disk of convergence $D_k$ the identity theorem applies, and the function extends uniquely along the path $\gamma$.

3. The Monodromy Theorem

Definition (Continuation Along a Path)

Let $\gamma: [0,1] \to \mathbb{C}$ be a curve and let $f_0$ be holomorphic on a region $D_0$ containing $\gamma(0)$. The continuation of $f_0$ along $\gamma$ is a family of regions $(D_t)$ and functions $(f_t)$ such that for each $t$ we have $\gamma(t) \in D_t$, $f_t$ is holomorphic on $D_t$, and the functions agree on the overlaps of $D_t$ for nearby values of $t$. The value $f_1$ at the endpoint $\gamma(1)$ is the result of continuation along $\gamma$.

Theorem (Monodromy Theorem)

Let $D$ be a simply connected region and let $f$ be holomorphic in a neighborhood of a point $z_0 \in D$. If $f$ can be analytically continued along every path in $D$, then $f$ extends to a single-valued holomorphic function defined on all of $D$.

Idea of the Proof

Since $D$ is simply connected, any two curves with the same endpoints are homotopic (continuously deformable into each other). By the identity theorem, continuation along homotopic curves yields the same value. Hence the value at the endpoint is well-defined independently of the path, and $f$ is determined on all of $D$ as a single-valued function.

Conversely, in a non-simply-connected region the results of continuing along different paths may disagree. This is the origin of multivalued functions. For example, continuing $\log z$ over $\mathbb{C} \setminus \{0\}$ (non-simply-connected) shifts the value by $2\pi i$ each time one loops around the origin. How to handle this multivaluedness is the subject of the next section.

4. Multivalued Functions and Riemann Surfaces

Over a non-simply-connected region, analytic continuation is path dependent and produces multivalued functions. The stage on which this multivaluedness can be recast as a single-valued function is the Riemann surface. A Riemann surface is a covering space built by gluing several "sheets" at branch points so that the function becomes single-valued.

Example: The Logarithm

Continuing $f(z) = \log z$ along a curve that loops once around the origin from $z = 1$ adds $2\pi i$ to the original value. Because $\mathbb{C} \setminus \{0\}$ is not simply connected (a curve enclosing the origin cannot be contracted), the monodromy theorem does not apply. To treat $\log z$ as single-valued, one considers it on the universal covering space (an infinitely sheeted spiral surface) of $\mathbb{C} \setminus \{0\}$.

Example: The Square Root

For $f(z) = \sqrt{z}$, looping once around the origin turns $f$ into $-f$, and looping twice returns it to its original value. The Riemann surface that makes this single-valued is a double cover formed by joining two copies of the complex plane at the origin (the branch point). The two sheets swap around the branch points $z = 0, \infty$.

5. Representative Examples

5.1 The Riemann Zeta Function

The Dirichlet series $\zeta(s) = \displaystyle\sum_{n=1}^{\infty} n^{-s}$ converges only for $\mathrm{Re}(s) > 1$. Analytic continuation, however, extends it to all of $\mathbb{C}$ except for a simple pole at $s = 1$.

One method of continuation uses the alternating zeta function (the Dirichlet eta function)

$$\eta(s) = \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n-1}}{n^s} = (1 - 2^{1-s})\zeta(s)$$

which converges for $\mathrm{Re}(s) > 0$. The functional equation

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

then extends it to $\mathrm{Re}(s) < 0$ as well.

5.2 The Gamma Function

$\Gamma(s) = \displaystyle\int_0^{\infty} t^{s-1}e^{-t}\,dt$ is defined for $\mathrm{Re}(s) > 0$ (definition via an integral representation). Repeatedly applying the functional equation $\Gamma(s+1) = s\,\Gamma(s)$, that is $\Gamma(s) = \Gamma(s+1)/s$, continues it to all of $\mathbb{C}$ except for simple poles at $s = 0, -1, -2, \ldots$. This is a typical case of continuation combining an integral representation with a functional equation.

Figure 2. The idea of analytic continuation of the Riemann zeta function $\zeta(s)$. From the domain $\mathrm{Re}(s) > 1$ of the Dirichlet series, the eta function and the functional equation extend it to all of $\mathbb{C} \setminus \{1\}$. The nontrivial zeros relevant to the Riemann hypothesis lie in the critical strip $0 < \mathrm{Re}(s) < 1$.

6. Natural Boundaries

Not every function can be analytically continued. Sometimes the boundary of the disk of convergence is the very "edge" of the function.

Definition (Natural Boundary)

The boundary $\partial D$ of the domain $D$ of a holomorphic function $f$ is a natural boundary if $f$ cannot be analytically continued beyond $D$ in any direction.

Example: A Lacunary Series

The series $f(z) = \displaystyle\sum_{n=0}^{\infty} z^{2^n} = z + z^2 + z^4 + z^8 + \cdots$ converges for $|z| < 1$. Every point on the unit circle $|z| = 1$ is a singularity, so the unit circle is a natural boundary. Hence this function cannot be continued at all beyond the unit disk.

Theorem (Hadamard's Gap Theorem)

If, in a power series $\displaystyle\sum_{k=0}^{\infty} a_k z^{n_k}$, the exponent sequence $\{n_k\}$ satisfies the Hadamard gap condition

$$\dfrac{n_{k+1}}{n_k} \geq q > 1$$

then its disk of convergence is a natural boundary. The lacunary series above satisfies this condition with $n_k = 2^k$ (so $q = 2$).

7. The Schwarz Reflection Principle

Theorem (Schwarz Reflection Principle)

Let $D$ be a region in the upper half-plane whose boundary contains an open interval $I$ of the real axis. If $f$ is holomorphic on $D$ and takes real values on $I$, then $f$ can be analytically continued to the reflection $D^*$ of $D$ across the real axis, and on $D^*$ it satisfies $f(z) = \overline{f(\overline{z})}$.

The reflection principle uses a boundary condition on the real axis to "reflect" a holomorphic function from the upper half-plane to the lower half-plane. It plays an important role in constructing conformal maps and in extending the Riemann mapping theorem to the boundary.

8. Applications

8.1 Defining Special Functions

Many important functions (the gamma function, the zeta function, hypergeometric functions, and so on) are first defined on a limited region by an integral or a power series, and are then extended to a larger region by analytic continuation. Analytic continuation is also the means by which the "official domain" of a special function is fixed.

8.2 Evaluating Integrals and Parameters

The value of an integral or series depending on a parameter may be easy to compute in one region but divergent in another. Analytic continuation lets one "extend" a result obtained in a region where computation is easy to general parameter values, regularizing it (zeta-function regularization is based on this idea).

8.3 Applications to Number Theory

The analytic continuation of zeta and L-functions plays a fundamental role in studying the distribution of primes and number-theoretic conjectures (including the Riemann hypothesis). The behavior of the zeros in the critical strip can only be discussed on the analytically continued function.

9. Frequently Asked Questions

Q1. What is analytic continuation?

It is a technique for extending a holomorphic function defined on some region to a larger region while preserving holomorphy. By the identity theorem the extension is unique, and the basic method is to chain overlapping disks of convergence of power series.

Q2. What is the monodromy theorem?

It guarantees that analytic continuation within a simply connected region is path independent (extends to a single-valued function). In non-simply-connected regions path dependence arises and multivalued functions appear.

Q3. How are multivalued functions related to analytic continuation?

$\log z$ and $z^{1/2}$ change value when continued along a path encircling a branch point, even after returning to the start: $\log z$ shifts by $2\pi i$ per loop, and $\sqrt{z}$ changes sign after one loop. Viewed on a covering space called a Riemann surface, they become single-valued holomorphic functions.

Q4. What does analytic continuation of the zeta function mean?

$\zeta(s) = \displaystyle\sum n^{-s}$ converges only for $\mathrm{Re}(s) > 1$, but the eta function and the functional equation extend it to all of $\mathbb{C}$ except for a simple pole at $s = 1$. The Riemann hypothesis is the conjecture that all nontrivial zeros of this extended function lie on $\mathrm{Re}(s) = 1/2$.

10. References

Wikipedia, "Analytic continuation"
Wikipedia, "Monodromy theorem"
Wikipedia, "Lacunary function" (natural boundaries)
Wikipedia, "Schwarz reflection principle"
L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, 1979.
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., Oxford University Press, 1986.