Analytic Continuation

Analytic Continuation

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Published: 2026-03-20 · Updated: 2026-10-01

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 $D_2$. This is an extension to a single region containing $D_1$; continuation along a path by chaining disks is treated in Sections 2 and 3.

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 extend beyond the disk. Analytic continuation provides a way to extend a function beyond any single series representation. Note, however, that the uniqueness in the corollary above concerns a target region $D_2$ that is a connected region containing $D_1$. If one chains disks and moves away from $D_1$, different paths can lead to different functions (monodromy), so a single-valued global function need not exist. This 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).

  1. Start from the disk of convergence $D_0$ (of radius $R_0$) of a power series $\displaystyle\sum a_n (z - z_0)^n$ centered at $z_0$.
  2. Re-expand the Taylor series at another point $z_1 \in D_0$ to obtain a new disk of convergence $D_1$.
  3. If $D_1$ extends beyond $D_0$, the function is extended to $D_0 \cup D_1$.
  4. Repeat this procedure, advancing the continuation along a path.

Concretely, the re-expansion in the second step is

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

The radius of convergence $r_1$ of this new series is always at least $R_0 - |z_1 - z_0|$ (the distance from $z_1$ to the boundary of $D_0$). If $r_1 > R_0 - |z_1 - z_0|$, then $D_1$ sticks out of $D_0$, and $f$ has been extended beyond the original disk of convergence.

z₀ z₁ z₂ z₃ path of continuation γ D₀ D₁ D₂ D₃ agree on overlaps analytic continuation via a chain of disks
Figure 1. Analytic continuation via a chain of power-series disks. Each next center $z_{k+1}$ is taken inside the previous disk of convergence $D_k$, and on the overlaps the identity theorem makes the functions agree. Along this chain of disks the function is determined step by step (following a different path need not give the same result).

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 function $f_1$ obtained near the endpoint $\gamma(1)$ (the function element at $\gamma(1)$) 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 $D$ starting at $z_0$, 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 hypothesis, continuation is possible along every intermediate curve of the homotopy, so by the identity theorem the result does not change when the curve is moved slightly. Hence continuation along homotopic curves yields the same result, and 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, if a branch of $\log z$ chosen near $z=1$ is continued along paths in $\mathbb{C} \setminus \{0\}$ (non-simply-connected), its value shifts by $2\pi i$ each time one loops once around the origin in the positive (counterclockwise) direction. 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 can, for some functions, depend on the path and produce 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 one-dimensional complex manifold; to make a multivalued function single-valued, it is often constructed by gluing several "sheets" along cuts and adding the branch points where needed (a branched covering).

Example: The Logarithm

Continuing a branch of $f(z) = \log z$ chosen near $z = 1$ along a curve that loops once around the origin from $z = 1$ in the positive direction 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 two-sheeted branched covering, formed by cutting two copies of the complex plane along a ray from the origin (for example, the negative real axis), gluing them crosswise along the cut, and adding 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 and is holomorphic for $\mathrm{Re}(s) > 0$ (the identity on the right first holds for $\mathrm{Re}(s) > 1$). Setting $\zeta(s) = \eta(s)/(1 - 2^{1-s})$ extends $\zeta$ to $\mathrm{Re}(s) > 0$. At the points $s = 1 + 2\pi i k/\ln 2$ ($k \neq 0$) where the denominator vanishes, $\eta$ vanishes too and the singularity is removable, so the only remaining pole is $s = 1$. 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 $\mathrm{Re}(s) \le 0$ as well (the factor $\zeta(1-s)$ on the right is already known because $\mathrm{Re}(1-s) \ge 1$; at $s = 0$ the zero of $\sin$ cancels the pole at $1-s = 1$).

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.

Re(s) Im(s) Re(s) > 1 Dirichlet series σ = 1 σ = 0 extended by η(s) extended by functional eq. critical strip s=1 (pole) σ = 1/2
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 $\mathbb{C} \setminus \{1\}$. All nontrivial zeros of $\zeta$ lie in the critical strip $0 < \mathrm{Re}(s) < 1$, and the Riemann hypothesis asserts that they all lie on the critical line $\mathrm{Re}(s) = 1/2$.

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 for every point $p$ of $\partial D$, $f$ cannot be extended holomorphically to any neighborhood of $p$ (no boundary point admits an analytic continuation beyond $D$).

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 a power series $\displaystyle\sum_{k=0}^{\infty} a_k z^{n_k}$ ($a_k \neq 0$) has radius of convergence $R$ with $0 < R < \infty$, and its exponent sequence $\{n_k\}$ satisfies the Hadamard gap condition

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

then the circle $|z| = R$ 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 $D$, continuous on $D \cup I$, and takes real values on $I$, then, with $D^*$ the reflection of $D$ across the real axis, the function $F$ defined by $F = f$ on $D$ and $F(z) = \overline{f(\overline{z})}$ on $D^*$ (taking the values of $f$ on $I$) is holomorphic on $D \cup I \cup D^*$. That is, $f$ can be analytically continued to $D \cup I \cup D^*$.

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 of revealing how far a special function naturally extends, including its poles and branching; for functions that become multivalued, a choice of branch or Riemann surface must be made as well.

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. When analytic continuation is possible, it lets one "extend" a result obtained in a region where computation is easy to other parameter values. Regularization that assigns values to divergent quantities (such as zeta-function regularization) is based on this idea, but assigning values at poles and similar points requires additional conventions.

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, an extension to a connected region containing the original one is unique if it exists. When one continues along a path by chaining disks, however, different paths can lead to different branches. The basic method is to chain overlapping disks of convergence of power series.

Q2. What is the monodromy theorem?

It is the theorem that if $f$ can be analytically continued along every path in a simply connected region $D$, then the result does not depend on the path and $f$ extends to a single-valued holomorphic function on $D$ (continuations along two paths that are homotopic with fixed endpoints agree). In regions that are not simply connected, the result can depend on the path for some functions, 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 positive (counterclockwise) loop, and $\sqrt{z}$ changes sign after one loop. Viewed on 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 extends it to $\mathrm{Re}(s) > 0$ (the zeros of the denominator other than $s = 1$ are removable) and the functional equation to $\mathrm{Re}(s) \le 0$, giving 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