Analytic Continuation
Permanence of Functional Relations
Elementary Functions Properties
Analytic Continuation and Riemann Surfaces
Analytic Continuation Across a Boundary
Example: Geometric Series Continuation
The uniqueness theorem (here) guarantee that allows for the extension of an analytic function’s domain of definition in a well-defined manner. This process, known as analytic continuation, is a central concept in complex analysis.
We begin with a function element, which is a pair (f, \mathcal D), where \mathcal D is a domain and f(z) is a function analytic in \mathcal D. Often, the initial function element is given by a power series and its disk of convergence.
Consider two function elements, (f_1, \mathcal D_1) and (f_2, \mathcal D_2). We say that (f_2, \mathcal D_2) is a direct analytic continuation of (f_1, \mathcal D_1) if the intersection of their domains \mathcal D_1 \cap \mathcal D_2 is a non-empty domain, and if f_1(z) = f_2(z) for all z \in \mathcal D_1 \cap \mathcal D_2.
The uniqueness theorem ensures that such a continuation, if it exists, is unique. If we find any analytic function f_2 in \mathcal D_2 that agrees with f_1 on the intersection, we are certain it is the only possible one. Without this theorem, the concept of continuation would be ambiguous, as multiple different analytic functions in \mathcal D_2 could conceivably agree with f_1 on the shared domain.
The standard method of analytic continuation proceeds by constructing a chain of function elements. Suppose we start with a function element (f_0, \mathcal D_0), where \mathcal D_0 is the disk of convergence for a Taylor series centered at z_0.
f_0(z) = \sum_{n=0}^\infty c_n (z - z_0)^n, \quad z \in \mathcal D_0
We can choose a point z_1 \in \mathcal D_0, distinct from z_0. Since f_0(z) is analytic at z_1, we can compute its derivatives f_0^{(n)}(z_1) for all n. We then form a new Taylor series centered at z_1:
f_1(z) = \sum_{n=0}^\infty \frac{f_0^{(n)}(z_1)}{n!} (z - z_1)^n
This new series converges in some disk \mathcal D_1 centered at z_1. The intersection \mathcal D_0 \cap \mathcal D_1 is a non-empty open set. Within this intersection, both f_0(z) and f_1(z) are analytic and, by construction of the Taylor series, f_0(z) = f_1(z). The function element (f_1, \mathcal D_1) is the direct analytic continuation of (f_0, \mathcal D_0). The domain of our function has now been extended to \mathcal D_0 \cup \mathcal D_1.
This process can be repeated, generating a sequence of function elements (f_0, \mathcal D_0), (f_1, \mathcal D_1), (f_2, \mathcal D_2), \dots that forms a chain of analytic continuations along a path. The collection of all such function elements that can be obtained from one another through chains of direct analytic continuations constitutes a global analytic function.
A significant issue arises when the underlying domain is not simply connected. The result of analytic continuation between two points a and b can depend on the path taken. If we continue a function element along two different paths from a to b, the resulting function elements at b may not be identical.
This phenomenon gives rise to multi-valued functions, such as the complex logarithm or the square root function. The global analytic function is then properly defined not on the complex plane \mathbb C, but on a more elaborate geometric structure known as a Riemann surface, on which it is single-valued.
Real axis continuation
Let there be given, on an interval [a, b] of the real x-axis, a continuous function f(x) of a real variable.
If there exists an analytic function F(z) defined in a domain \mathcal D of the complex plane, where \mathcal D contains the interval [a, b], such that F(x) = f(x) for all x \in [a, b], then this function F(z) is the unique analytic continuation of f(x) into the domain \mathcal D.
The uniqueness follows directly from the fact that any two such functions would agree on the interval [a,b], which contains a sequence of points with a limit point in \mathcal D.
We can extend elementary functions of a real variable, in particular those that can be represented by a Taylor series.
Consider the exponential and trigonometric functions, which possess Taylor series expansions that converge for all x \in \mathbb{R}:
\begin{aligned} & e^x = \sum_{n=0}^\infty \frac{x^n}{n!} \\ & \sin(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} \\ & \cos(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!} \end{aligned}
We can formally define complex functions by substituting the real variable x with the complex variable z in these series:
\begin{aligned} & \sum_{n=0}^\infty \frac{z^n}{n!}\\ & \sum_{n=0}^\infty (-1)^n \frac{z^{2n+1}}{(2n+1)!} \\ & \sum_{n=0}^\infty (-1)^n \frac{z^{2n}}{(2n)!} \end{aligned}
These power series converge for all z \in \mathbb{C}, which can be verified using the ratio test or the Cauchy-Hadamard formula. Consequently, they define functions that are analytic over the entire complex plane; such functions are known as entire functions.
Since these entire functions coincide with their real counterparts on the real axis, the uniqueness theorem dictates that they are the unique analytic continuations of e^x, \sin(x), and \cos(x) into the complex plane.
It is therefore natural to define the complex exponential and trigonometric functions by these series:
\begin{aligned} & e^z \equiv \sum_{n=0}^\infty \frac{z^n}{n!} \\ & \sin(z) \equiv \sum_{n=0}^\infty (-1)^n \frac{z^{2n+1}}{(2n+1)!} \\ & \cos(z) \equiv \sum_{n=0}^\infty (-1)^n \frac{z^{2n}}{(2n)!} \end{aligned}
From the complex exponential function, we can define the hyperbolic functions for a complex variable z:
\begin{aligned} \cosh(z) &= \frac{e^z + e^{-z}}{2} \\ \sinh(z) &= \frac{e^z - e^{-z}}{2} \end{aligned}
As sums and compositions of entire functions, these are also entire functions. Other trigonometric functions like \tan(z) = \sin(z)/\cos(z) can be defined analogously. These functions are not entire, as they are not analytic at the points where their denominators vanish.
Now consider the natural logarithm. The Taylor series for \ln(x) around x=1 is given by:
\ln(x) = \sum_{n=1}^\infty (-1)^{n-1}\frac{(x-1)^n}{n}
This series converges on the real interval (0, 2]. We extend this to the complex domain by considering the series:
\sum_{n=1}^\infty (-1)^{n-1}\frac{(z-1)^n}{n}
The radius of convergence of this series is R=1. Therefore, it defines an analytic function in the open disk |z-1|<1. We define the principal branch of the complex logarithm in this disk by this series:
\ln(z) \equiv \sum_{n=1}^\infty (-1)^{n-1}\frac{(z-1)^n}{n}
This function is the unique analytic continuation of \ln(x) from the interval (0,2) into the disk |z-1|<1. It can be shown that this series represents the same function as the one defined by the path integral:
\ln(z) = \int_1^z \frac{1}{\zeta} \, \mathrm d\zeta
provided the path of integration lies within a simply connected domain where 1/\zeta is analytic and which contains the disk |z-1|<1.
The process illustrated above can be generalized. If a function f(x) of a real variable x is represented by a power series:
f(x) = \sum_{n=0}^\infty c_n (x - x_0)^n
convergent on an interval (x_0-R, x_0+R), then the complex power series:
F(z) = \sum_{n=0}^\infty c_n (z - z_0)^n
defines the unique analytic continuation of f(x) into the open disk |z - z_0| < R. Furthermore, a direct consequence of the properties of power series is that the derivative F^{(k)}(z) of the continued function is the unique analytic continuation of the real derivative f^{(k)}(x).
Permanence of functional relations
The uniqueness theorem for analytic functions has, as consequence, that functional identities that hold for real variables often remain valid when those variables are extended into the complex domain.
This principle allows to use the knowledge of real analysis to establish fundamental properties of complex functions.
We can demonstrate this principle for relations involving a single variable and for those involving multiple variables.
To illustrate, consider the fundamental trigonometric identity for a real variable x:
\sin^2(x) + \cos^2(x) = 1
To determine if this relation holds for a complex variable z, we can define an auxiliary function:
F(z) = \sin^2(z) + \cos^2(z) - 1
The functions \sin(z) and \cos(z) are entire functions, meaning they are analytic throughout the complex plane. Since sums and products of analytic functions are analytic, F(z) is also an entire function. On the real axis (z=x), we know that F(x) = \sin^2(x) + \cos^2(x) - 1 = 0. The real axis contains a sequence of points with a limit point within the domain of analyticity of F(z) (which is all of \mathbb{C}).
By the uniqueness theorem, if an analytic function is zero on such a set, it must be identically zero throughout its domain and therefore:
F(z) = \sin^2(z) + \cos^2(z) - 1 \equiv 0
This confirms that the identity holds for all z \in \mathbb{C}. This approach can be generalized into a formal theorem.
Single variable functions
Theorem: Let F(w_1, \dots, w_n) be a function of the complex variables w_1, \dots, w_n, analytic with respect to each variable w_i in its respective domain \mathcal D_i. Let f_1(z), \dots, f_n(z) be n functions of a single complex variable z, each analytic in a domain \mathcal D such that for every z \in \mathcal D, the value f_i(z) lies in \mathcal D_i. If the domain \mathcal D contains an interval [a,b] of the real axis and the relation F[f_1(x), \dots, f_n(x)]=0 holds for all x \in [a,b], then F[f_1(z), \dots, f_n(z)]=0 for all z \in \mathcal D.
Proof: The proof hinges on showing that the composite function \Phi(z) = F[f_1(z), \dots, f_n(z)] is analytic in \mathcal D. If this is established, since \Phi(x) = 0 for x \in [a,b], the uniqueness theorem directly implies that \Phi(z) \equiv 0 in \mathcal D.
To demonstrate that \Phi(z) is analytic, it suffices to show that its complex derivative exists at every point in \mathcal D. For simplicity, we demonstrate this for two variables; the generalization to n variables follows the same pattern. Let:
\Phi(z) = F[f_1(z), f_2(z)]
Let z_0 be an arbitrary point in \mathcal D. The derivative of \Phi(z) at z_0 is given by the limit:
\Phi^\prime(z_0) = \lim_{\Delta z \to 0} \frac{\Phi(z_0 + \Delta z) - \Phi(z_0)}{\Delta z}
We fix an arbitrary point z_0 \in \mathcal D and set f_1(z_0) = w_1^0 and f_2(z_0) = w_2^0. Let’s consider the expression:
\Phi(z_0 + \Delta z) - \Phi(z_0) = F[w_1^0 + \Delta w_1, w_2^0 + \Delta w_2] - F[w_1^0, w_2^0]
\Delta w_1 and \Delta w_2 are the increment of the function f_1(z) and f_2(z) due to the increment \Delta z of the independent variable z. By hypothesis, the partial derivatives are existing and they are continuous, applying the chain rule:
\begin{aligned} \Phi(z_0 + \Delta z) - \Phi(z_0) = & F[w_1^0, w_2^0] + \frac{\partial F}{\partial w_1}F[w_1^0, w_2^0 + \Delta w_2]\Delta w_1 + \eta_1 \Delta w_1 + \\ & + \frac{\partial F}{\partial w_2}F[w_1^0 + \Delta w_1, w_2^0]\Delta w_2 + \eta_2 \Delta w_2 - F[w_1^0, w_2^0] \end{aligned}
\eta_1 and \eta_2 are infinitesimal for \Delta w_1 and \Delta w_2 so they go to zero as z \to 0. Diving by \Delta z and passing to the limit for \Delta z \to 0:
\begin{aligned} \lim {\Delta z \to 0}\frac{\Phi(z_0 + \Delta z) - \Phi(z_0)}{\Delta z} = & \frac{\partial F}{\partial w_1}F[w_1^0, w_2^0] \frac{\Delta w_1}{\Delta z} + \frac{\partial F}{\partial w_2}F[w_1^0, w_2^0]\frac{\Delta w_2}{\Delta z} \\ = & \frac{\partial F}{\partial w_1}F[w_1^0, w_2^0] \frac{\Delta f_1(z_0)}{\Delta z} + \frac{\partial F}{\partial w_2}F[w_1^0, w_2^0]\frac{\Delta f_2(z_0)}{\Delta z} \\ = & \frac{\partial F}{\partial w_1}F[w_1^0, w_2^0] f_1^\prime(z_0) + \frac{\partial F}{\partial w_2}F[w_1^0, w_2^0]f_2^\prime(z_0) \end{aligned}
Then the derivative exists and is given by:
\Phi^\prime(z_0) = \frac{\partial F}{\partial w_1}\bigg|_{(f_1(z_0), f_2(z_0))} f_1^\prime(z_0) + \frac{\partial F}{\partial w_2}\bigg|_{(f_1(z_0), f_2(z_0))} f_2^\prime(z_0)
The existence of the partial derivatives of F and the derivatives of f_1 and f_2 is guaranteed by their analyticity. Since the expression for \Phi^\prime(z_0) is a combination of continuous functions, the derivative \Phi^\prime(z) is continuous in \mathcal D. The existence of a continuous derivative confirms that \Phi(z) is analytic in \mathcal D.
Several variables functions
The principle can be extended to relations involving functions of multiple, independent complex variables.
Theorem: Let f_1(z_1), \dots, f_n(z_n) be functions analytic in domains \mathcal D_1, \dots, \mathcal D_n respectively. Assume each domain \mathcal D_i contains a real interval [a_i, b_i]. Let F(w_1, \dots, w_n) be a function analytic in its arguments over the ranges of the functions f_i. If the relation F[f_1(x_1), \dots, f_n(x_n)]=0 holds for all real x_i \in [a_i, b_i], then the relation F[f_1(z_1), \dots, f_n(z_n)]=0 holds for all complex z_i \in \mathcal D_i.
Proof: The proof proceeds by an iterative application of the uniqueness theorem for a single variable.
First, fix the real variables x_2, \dots, x_n to arbitrary values x_2^0 \in [a_2, b_2], \dots, x_n^0 \in [a_n, b_n]. Consider the function of a single complex variable z_1:
\Phi_1(z_1) = F[f_1(z_1), f_2(x_2^0), \dots, f_n(x_n^0)]
This function is analytic in z_1 \in \mathcal D_1. By hypothesis, \Phi_1(x_1) = 0 for all x_1 \in [a_1, b_1]. By the uniqueness theorem, \Phi_1(z_1) \equiv 0 for all z_1 \in \mathcal D_1. Since the choices for x_2^0, \dots, x_n^0 were arbitrary, we have established that the relation holds for a complex first variable and real subsequent variables: F[f_1(z_1), f_2(x_2), \dots, f_n(x_n)]=0.
Next, fix z_1 to an arbitrary value z_1^0 \in \mathcal D_1 and the real variables x_3, \dots, x_n to arbitrary values. Consider the function of the complex variable z_2:
\Phi_2(z_2) = F[f_1(z_1^0), f_2(z_2), f_3(x_3^0), \dots, f_n(x_n^0)]
This function is analytic in z_2 \in \mathcal D_2. From the previous step, we know that \Phi_2(x_2) = 0 for all x_2 \in [a_2, b_2]. The uniqueness theorem again implies that \Phi_2(z_2) \equiv 0 for all z_2 \in \mathcal D_2. As z_1^0 was arbitrary, the relation now holds for complex z_1, z_2 and real x_3, \dots, x_n.
We continue this process for each variable z_i in sequence. After n steps, we conclude that the relation F[f_1(z_1), \dots, f_n(z_n)]=0 holds for all z_i \in \mathcal D_i, thereby proving the theorem.
As an example, consider the addition formula for the exponential function, which is known to hold for real variables: e^{x_1} e^{x_2} = e^{x_1 + x_2}. We can prove it for complex variables z_1, z_2 by defining the function:
F(z_1, z_2) = e^{z_1} e^{z_2} - e^{z_1+z_2}
This function is entire in both z_1 and z_2. Fix z_2=y to be a real number. Then F(z_1, y) = e^{z_1} e^y - e^{z_1+y} is an entire function of z_1.
We know F(x,y)=0 for all real x. By the uniqueness theorem, F(z_1, y)=0 for all complex z_1 and real y.
Now, fix z_1=z_1^0 to be an arbitrary complex number. The function F(z_1^0, z_2) = e^{z_1^0} e^{z_2} - e^{z_1^0+z_2} is an entire function of z_2.
We know from the previous step that it is zero for all real z_2=y.
the uniqueness theorem implies it is zero for all complex z_2. Since z_1^0 was arbitrary, the relation holds for all z_1, z_2 \in \mathbb{C}.
Elementary functions properties
The power series representations of elementary functions provide a framework for extending their domains from the real line to the entire complex plane.
We begin with the series for the exponential function, which is absolutely convergent for all z \in \mathbb{C}:
e^z = \sum_{n=0}^\infty \frac{z^n}{n!}
By substituting z with iz, we can establish a connection to the trigonometric functions:
\begin{aligned} e^{iz} & = \sum_{n=0}^\infty \frac{(iz)^n}{n!} \\ & = \sum_{k=0}^\infty \frac{(iz)^{2k}}{(2k)!} + \sum_{k=0}^\infty \frac{(iz)^{2k+1}}{(2k+1)!} \\ & = \sum_{k=0}^\infty \frac{i^{2k}z^{2k}}{(2k)!} + \sum_{k=0}^\infty \frac{i^{2k+1}z^{2k+1}}{(2k+1)!} \\ & = \sum_{k=0}^\infty \frac{(-1)^k z^{2k}}{(2k)!} + i \sum_{k=0}^\infty \frac{(-1)^k z^{2k+1}}{(2k+1)!} \end{aligned}
The two series on the final line are the series for \cos(z) and \sin(z), respectively. This yields Euler’s formula, an identity valid for all complex numbers z:
e^{iz} = \cos(z) + i\sin(z)
This formula allows for the expression of trigonometric functions in terms of the exponential function.
By considering Euler’s formula for both z and -z, and using the even and odd properties of cosine and sine, we obtain a system of two linear equations for \cos(z) and \sin(z):
\begin{aligned} & e^{iz} = \cos(z) + i\sin(z) \\ & e^{-iz} = \cos(z) - i\sin(z) \end{aligned}
Solving this system yields the following expressions:
\begin{aligned} & \cos(z) = \frac{e^{iz} + e^{-iz}}{2}\\ & \sin(z) = \frac{e^{iz} - e^{-iz}}{2i} \end{aligned}
These relations are the complex-variable generalizations of the definitions used in real analysis.
Furthermore, they reveal a direct correspondence with the hyperbolic functions. Recalling the definitions \cosh(w) = (e^w + e^{-w})/2 and \sinh(w) = (e^w - e^{-w})/2, we can set w = iz to find:
\begin{aligned} & \cos(z) = \cosh(iz)\\ & \sin(z) = -i\sinh(iz) \end{aligned}
These identities demonstrate that, in the complex plane, trigonometric and hyperbolic functions are rotations of one another in their argument.
The inverse of the exponential function is the complex logarithm.
To define it, we first analyze the mapping w = e^z. Let z = x + iy, where x, y \in \mathbb{R}. The mapping becomes:
w = e^{x+iy} = e^x e^{iy}
Applying Euler’s formula to the e^{iy} term gives the polar representation of w:
w = e^x(\cos(y) + i\sin(y))
From this form, we can identify the modulus and argument of w:
|w| = e^x, \qquad \arg(w) = y + 2k\pi, \quad k \in \mathbb{Z}
The periodicity of the exponential function, e^{z + 2ik\pi} = e^z, implies that its inverse, the logarithm, is inherently multi-valued.
To find the expression for z in terms of w, we solve for x and y:
x = \ln(|w|), \qquad y = \arg(w)
Here, \ln(|w|) is the standard real-valued natural logarithm, since |w| is a positive real number.
Substituting these back into z=x+iy, we obtain the general definition of the complex logarithm:
\ln(w) = \ln(|w|) + i\arg(w)
Due to the 2k\pi ambiguity in the argument, for any non-zero complex number w, there are infinitely many values for \ln(w).
To work with a single-valued function, we define the principal value of the logarithm, denoted \operatorname{Ln}(z), by restricting the argument to the interval (-\pi, \pi].
This choice necessitates a branch cut, conventionally placed along the negative real axis, where the function is discontinuous.
On the domain \mathbb{C} \setminus (-\infty, 0], the principal value \operatorname{Ln}(z) is an analytic function and serves as the inverse to the exponential function, meaning e^{\operatorname{Ln}(z)} = z.
As a computational example, let us determine the logarithm of z = 1+i. First, we compute its modulus and principal argument:
\begin{aligned} & |1+i| = \sqrt{1^2 + 1^2} = \sqrt{2} \\ & \mathrm{Arg}(1+i) = \frac{\pi}{4} \end{aligned}
The principal value of the logarithm is:
\operatorname{Ln}(1+i) = \ln(\sqrt{2}) + i\frac{\pi}{4} = \frac{1}{2}\ln(2) + i\frac{\pi}{4}
The set of all possible values for the logarithm is given by including the multi-valued nature of the argument:
\ln(1+i) = \frac{1}{2}\ln(2) + i\left(\frac{\pi}{4} + 2k\pi\right), \quad k \in \mathbb{Z}
Analytic continuation and Riemann surfaces
As mentioned, the process of analytic continuation is a method to extend the domain of definition of a given analytic function. This procedure is founded on the uniqueness theorem for analytic functions, which ensures that such an extension, when possible, is uniquely determined.
Let us begin with a function element (f_1, \mathcal D_1), which consists of a function f_1(z) that is analytic in a domain \mathcal D_1.
Suppose there exists another domain \mathcal D_2 that has a non-empty intersection with \mathcal D_1, forming a connected subdomain \mathcal D_{12} = \mathcal D_1 \cap \mathcal D_2.
If we can find a function f_2(z), analytic in \mathcal D_2, such that f_1(z) = f_2(z) for all z \in \mathcal D_{12}, then f_2(z) is called the direct analytic continuation of f_1(z) into \mathcal D_2.
By the uniqueness theorem, if such an analytic function f_2(z) exists, it is the only one. Consequently, we can define a single analytic function F(z) on the larger domain \mathcal D = \mathcal D_1 \cup \mathcal D_2 as follows:
F(z) = \begin{cases} f_1(z), & \quad z \in \mathcal D_1 \\ f_2(z), & \quad z \in \mathcal D_2 \end{cases}
This function F(z) is well-defined and analytic throughout \mathcal D. We say that F(z) is the analytic continuation of f_1(z) into \mathcal D, and (F, \mathcal D) is a larger function element.
A more complex situation arises when the process of analytic continuation depends on the path taken. This can occur when the intersection of domains is not a single connected region. Consider two domains, \mathcal D_1 and \mathcal D_2, whose intersection \mathcal D_{12} consists of two disjoint regions, \mathcal D_{12}^\prime and \mathcal D_{12}^{\prime\prime}.
Let f_1(z) be analytic in \mathcal D_1 and f_2(z) be analytic in \mathcal D_2. Suppose that these two functions coincide in one of the intersection regions, say f_1(z) = f_2(z) for z \in \mathcal D_{12}^\prime, but they differ in the other region, f_1(z) \neq f_2(z) for z \in \mathcal D_{12}^{\prime\prime}.
In this scenario, we cannot define a single-valued analytic function on the union \mathcal D = \mathcal D_1 \cup \mathcal D_2. Attempting to do so leads to an ambiguity for any point z_0 \in \mathcal D_{12}^{\prime\prime}, where the function would need to assume two different values simultaneously, f_1(z_0) and f_2(z_0). This gives rise to what is termed a multi-valued analytic function.
We can define a single-valued function F^\prime(z) on the domain \mathcal D^\prime = (\mathcal D_1 \cup \mathcal D_2) \setminus \mathcal D_{12}^{\prime\prime}. However, from this starting point, we can continue F^\prime(z) into the region \mathcal D_{12}^{\prime\prime} in two distinct ways, yielding two different analytic functions on the full domain \mathcal D = \mathcal D_1 \cup \mathcal D_2:
\begin{aligned} & F_1(z) = \begin{cases} f_1(z), & \quad z \in \mathcal D_1 \\ f_2(z), & \quad z \in \mathcal D_2 \setminus \mathcal D_{12}^{\prime\prime} \end{cases} \\ & F_2(z) = \begin{cases} f_2(z), & \quad z \in \mathcal D_2 \\ f_1(z), & \quad z \in \mathcal D_1 \setminus \mathcal D_{12}^{\prime\prime} \end{cases} \end{aligned}
The function F_1(z) is analytic in \mathcal D and corresponds to extending by f_1(z) into \mathcal D_{12}^{\prime\prime}, while F_2(z) is also analytic in \mathcal D but corresponds to extending by f_2(z).
At any point z_0 \in \mathcal D_{12}^{\prime\prime}, we have F_1(z_0) = f_1(z_0) and F_2(z_0) = f_2(z_0), which are different. The collection of these function elements, \{ (F_1, \mathcal D), (F_2, \mathcal D) \}, constitutes the global analytic function, which is multi-valued over \mathcal D_{12}^{\prime\prime}.
To resolve the issue of multi-valued function, Riemann introduced a geometric idea: instead of viewing the function as having multiple values over a single domain, we can construct a new, more elaborate domain upon which the function is single-valued.
This new domain is called a Riemann surface.
The construction begins by taking two separate copies of the domains, which we can think of as two sheets. One sheet corresponds to the function element (f_1, \mathcal D_1) and the other to (f_2, \mathcal D_2). These sheets are then “glued” together.
In the region \mathcal D_{12}^\prime, where f_1(z) = f_2(z), the two sheets are joined. A path that moves from \mathcal D_1 \setminus \mathcal D_2 into \mathcal D_2 \setminus \mathcal D_1 through the “gateway” \mathcal D_{12}^\prime transitions smoothly from the first sheet to the second.
However, in the region \mathcal D_{12}^{\prime\prime}, where f_1(z) \neq f_2(z), the sheets are not joined. They lie one “above” the other without intersecting. A point z_0 \in \mathcal D_{12}^{\prime\prime} in the complex plane is represented by two distinct points on this new manifold: one point on the first sheet and another on the second sheet.
On this newly constructed geometric manifold, the multi-valued function F(z) becomes a single-valued function. At the point corresponding to z_0 on the first sheet, the function’s value is f_1(z_0). At the point corresponding to z_0 on the second sheet, its value is f_2(z_0). The ambiguity is resolved because the points in the domain have been separated.
This manifold is the Riemann surface for the global analytic function generated by f_1(z) and f_2(z). The distinct single-valued functions (F_1(z) and F_2(z) in our example) are called branches of the multi-valued function, and each branch is defined on a corresponding sheet of the Riemann surface.
Analytic continuation across a boundary
The concept of analytic continuation can be formulated under specific conditions when two domains share a common boundary.
This scenario, called analytic continuation across a boundary, provides a method for “gluing” two analytic functions together into a single, larger analytic function.
Let us consider two disjoint domains, \mathcal D_1 and \mathcal D_2, in the complex plane whose boundaries share a common, piecewise smooth curve C_{12}. We can define a composite domain \mathcal D = \mathcal D_1 \cup \mathcal D_2 \cup C_{12}.
We can formulate the following theorem.
Theorem: suppose f_1(z) is a function analytic in \mathcal D_1 and continuous on its closure \mathcal D_1 \cup C_{12}. Similarly, let f_2(z) be analytic in \mathcal D_2 and continuous on its closure \mathcal D_2 \cup C_{12}. If these two functions coincide on their common boundary, such that f_1(z) = f_2(z) for all z \in C_{12}, then the function F(z) defined as
F(z) = \begin{cases} f_1(z), & \quad z \in \mathcal D_1 \cup C_{12} \\ f_2(z), & \quad z \in \mathcal D_2 \cup C_{12} \end{cases}
is analytic throughout the entire domain \mathcal D.
Proof: to establish the analyticity of F(z), we need to show that its complex derivative exists at every point in \mathcal D. The function is already analytic by definition in the open domains \mathcal D_1 and \mathcal D_2. The verification is necessary for any point z_0 on the common boundary C_{12}.
Let z_0 be an arbitrary point on C_{12}. We can construct an open disk centered at z_0 with a sufficiently small radius such that its boundary, a circle we denote C_0, is entirely contained within the combined domain \mathcal D. For any point z inside this circle C_0, we define a new function G(z) using a Cauchy-type integral:
G(z) = \frac{1}{2\pi i} \oint_{C_0} \frac{F(\zeta)}{\zeta-z} \, \mathrm d\zeta
A property of Cauchy-type integrals is that G(z) is an analytic function of z for all z inside the contour C_0. The objective is to demonstrate that this analytic function G(z) is identical to our original function F(z) within the disk bounded by C_0. If this identity holds, it implies that F(z) must be analytic in a neighborhood of z_0.
Let consider a circle C_0 centered at z_0 and contained within \mathcal D = \mathcal D_1 \cup \mathcal D_2 \cup C_{12}. We will demonstrate the identity F(z) = G(z) for a point z inside the disk bounded by C_0 and located within \mathcal D_1.
The contour of integration C_0 is composed of two open arcs, C_1 = C_0 \cap \mathcal D_1 and C_2 = C_0 \cap \mathcal D_2. We can split the integral defining G(z) into two parts:
G(z) = \frac{1}{2\pi i} \int_{C_1} \frac{F(\zeta)}{\zeta-z} \, \mathrm d\zeta + \frac{1}{2\pi i} \int_{C_2} \frac{F(\zeta)}{\zeta-z} \, \mathrm d\zeta
By the definition of F(z), we substitute F(\zeta) with f_1(\zeta) on C_1 and with f_2(\zeta) on C_2:
G(z) = \frac{1}{2\pi i} \int_{C_1} \frac{f_1(\zeta)}{\zeta-z} \, \mathrm d\zeta + \frac{1}{2\pi i} \int_{C_2} \frac{f_2(\zeta)}{\zeta-z} \, \mathrm d\zeta
Now, we will add and subtract an integral over the boundary segment \gamma_{12}. This manipulation allows us to form closed contours:
\begin{aligned} G(z) = & \left( \frac{1}{2\pi i} \int_{C_1} \frac{f_1(\zeta)}{\zeta-z} \, \mathrm d\zeta + \frac{1}{2\pi i} \int_{\gamma_{12}} \frac{f_1(\zeta)}{\zeta-z} \, \mathrm d\zeta \right) \\ & + \left( \frac{1}{2\pi i} \int_{C_2} \frac{f_2(\zeta)}{\zeta-z} \, \mathrm d\zeta - \frac{1}{2\pi i} \int_{\gamma_{12}} \frac{f_1(\zeta)}{\zeta-z} \, \mathrm d\zeta \right) \end{aligned}
Let us analyze the terms separately.
The first term combines the integral over the arc C_1 and the segment \gamma_{12} to form a closed contour, which we can call \Gamma_1 = C_1 \cup \gamma_{12}. Since our point z is inside the region enclosed by \Gamma_1, Cauchy’s integral formula applies directly:
\frac{1}{2\pi i} \oint_{\Gamma_1} \frac{f_1(\zeta)}{\zeta-z} \, \mathrm d\zeta = f_1(z)
Now we address the second term. Using the hypothesis f_1(\zeta) = f_2(\zeta) on the boundary segment \gamma_{12}, we can replace f_1 with f_2 in the integral:
\frac{1}{2\pi i} \int_{C_2} \frac{f_2(\zeta)}{\zeta-z} \, \mathrm d\zeta - \frac{1}{2\pi i} \int_{\gamma_{12}} \frac{f_2(\zeta)}{\zeta-z} \, \mathrm d\zeta
This expression can be recognized as an integral over another closed contour, \Gamma_2 = C_2 \cup (-\gamma_{12}), where -\gamma_{12} indicates traversal in the opposite direction. The expression is equivalent to:
\frac{1}{2\pi i} \oint_{\Gamma_2} \frac{f_2(\zeta)}{\zeta-z} \, \mathrm d\zeta
The function f_2(\zeta) is analytic within the region enclosed by \Gamma_2. The point z lies in \mathcal D_1, which is outside the region enclosed by \Gamma_2. Therefore, the integrand f_2(\zeta)/(\zeta-z) is analytic everywhere inside the contour \Gamma_2. By Cauchy’s integral theorem, this integral over a closed loop is zero:
\frac{1}{2\pi i} \oint_{\Gamma_2} \frac{f_2(\zeta)}{\zeta-z} \, \mathrm d\zeta = 0
Substituting these results back into our expression for G(z), we obtain:
G(z) = f_1(z) + 0 = f_1(z)
Since we assumed z \in \mathcal D_1, we have F(z) = f_1(z). We have therefore shown that G(z) = F(z) for any point z in the part of the disk within \mathcal D_1. A symmetric argument proves the same for any z in \mathcal D_2. Because F(z) and G(z) are continuous, they must be equal on \gamma_{12} as well.
The identity F(z) = G(z) holds throughout the neighborhood of z_0 defined by the disk. Since G(z) is analytic, F(z) must also be analytic in this neighborhood. As z_0 was an arbitrary point on the boundary, this completes the proof.
Example: n-th root extraction
Let us examine the function defined by the extraction of the n-th root of a complex number z. In polar coordinates, where z = r e^{i\theta} with r > 0, the operation yields multiple values.
The complex number w is an n-th root of z if w^n = z. This equation has n distinct solutions, given by:
w_k = \sqrt[n]{r} e^{i\frac{\theta + 2\pi k}{n}}, \quad k \in \{0, 1, \dots, n-1\}
This expression defines a multivalued function. To work with this function within the framework of complex analysis, we must select a single-valued, analytic branch. This is accomplished by restricting the domain of the argument \theta.
A common choice is to define a principal branch by constraining the argument. Let us adopt the convention 0 \le \arg(z) < 2\pi. This defines the principal branch of the root function as:
w_0(z) = \sqrt[n]{r} e^{i\frac{\theta}{n}}, \quad 0 \le \theta < 2\pi
For this function w_0(z) to be analytic, its domain must be an open set where it is continuous. The choice 0 \le \theta < 2\pi introduces a discontinuity along the positive real axis. A point z=x on this axis can be approached from the upper half-plane (\theta \to 0^+) or the lower half-plane (\theta \to 2\pi^-), yielding different results:
\begin{aligned} \lim_{\theta \to 0^+} w_0(re^{i\theta}) &= \sqrt[n]{r} \\ \lim_{\theta \to 2\pi^-} w_0(re^{i\theta}) &= \sqrt[n]{r} e^{i\frac{2\pi}{n}} \end{aligned}
To render w_0(z) analytic, we must remove the positive real axis from its domain. We define the first sheet, \mathcal{S}_0, as the complex plane with a branch cut along this axis.
The function w_0(z) maps the sheet \mathcal{S}_0 bijectively onto the sector in the w-plane defined by 0 \le \arg(w) < 2\pi/n.
Let us now consider a second function, w_1(z), corresponding to k=1:
w_1(z) = \sqrt[n]{r} e^{i\frac{\theta + 2\pi}{n}}
This function is naturally defined for a different range of the argument, for instance 2\pi \le \theta < 4\pi.
Its domain can be visualized as a second copy of the complex plane, which we will call sheet \mathcal{S}_1, also cut along the positive real axis. This function maps the sheet \mathcal{S}_1 onto the sector 2\pi/n \le \arg(w) < 4\pi/n.
The concept of analytic continuation allows us to connect these branches. Observe the limiting values of w_0 and w_1 on the boundaries of their respective cuts.
The value of w_0(z) as z approaches the cut from below (the “lower edge” of the cut on \mathcal{S}_0, where \arg(z) \to 2\pi^-) is \sqrt[n]{r} \exp(i 2\pi/n). The value of w_1(z) as z approaches the cut from above (the “upper edge” of the cut on \mathcal{S}_1, where \arg(z) \to 2\pi^+) is also \sqrt[n]{r} \exp(i 2\pi/n).
Because these limiting values coincide, we can define a single analytic function across the boundary. This is geometrically realized by identifying, or “gluing,” the lower edge of the branch cut on sheet \mathcal{S}_0 with the upper edge of the branch cut on sheet \mathcal{S}_1.
This process can be repeated for each value of k from 0 to n-1. For each k, we define a branch and a corresponding sheet:
w_k(z) = \sqrt[n]{r} e^{i\frac{\theta + 2\pi k}{n}}, \quad 2\pi k \le \theta < 2\pi(k+1)
The function w_k(z) is the analytic continuation of w_{k-1}(z) across their shared boundary. The lower edge of the cut on sheet \mathcal{S}_{k-1} is identified with the upper edge of the cut on sheet \mathcal{S}_k. This construction is performed for k = 1, \dots, n-1.
To complete the structure, we examine the final sheet, \mathcal{S}_{n-1}. As z approaches the lower edge of its cut, its argument approaches 2\pi n. The value of the function w_{n-1}(z) approaches:
\sqrt[n]{r} e^{i\frac{2\pi(n-1) + 2\pi}{n}}= \sqrt[n]{r} e^{i\frac{2\pi n}{n}} = \sqrt[n]{r} e^{i 2\pi} = \sqrt[n]{r}
This is the value of the function w_0(z) on the upper edge of the cut on the first sheet, \mathcal{S}_0. Consequently, the lower edge of the cut on the final sheet \mathcal{S}_{n-1} is identified with the upper edge of the cut on the first sheet \mathcal{S}_0.
This construction of n interconnected sheets forms a new manifold, the Riemann surface for the function w = z^{1/n}. On this surface, the function is single-valued and analytic everywhere except at special points. The function defined on this entire surface is called the complete analytic function.
A point is designated a branch point if analytic continuation along any sufficiently small closed curve encircling it results in a transition from one sheet of the Riemann surface to another.
For the function w = z^{1/n}, consider a small circular path z = \epsilon e^{i\phi} around the origin, with \phi varying from 0 to 2\pi. As z completes this circuit, its argument \theta increases by 2\pi.
The value of the function changes from \sqrt[n]{\epsilon} \exp(i\phi_0/n) to \sqrt[n]{\epsilon} \exp(i(\phi_0+2\pi)/n), which corresponds to moving from sheet \mathcal{S}_k to \mathcal{S}_{k+1} and therefore, z=0 is a branch point.
To analyze the point at infinity, we use the transformation z = 1/\zeta and examine the behavior near \zeta=0. The function becomes w = (1/\zeta)^{1/n} = \zeta^{-1/n}. A circuit around \zeta=0 also causes a transition between branches. It follows that z=\infty is also a branch point.
The two branch points for the function w = z^{1/n} are z=0 and z=\infty. Encircling any other point in the complex plane does not cause a change of sheet.
Power series continuation
An alternative method for constructing the analytic continuation of a function relies on the properties of power series. This approach builds the global function element by element, starting from a local representation.
Suppose a function f_1(z) is analytic in a given domain \mathcal D_1. We can select an arbitrary point z_0 within this domain and expand f_1(z) into its Taylor series centered at z_0:
f_1(z) = \sum_{n=0}^\infty c_n (z - z_0)^n = \sum_{n=0}^\infty \frac{f_1^{(n)}(z_0)}{n!} (z - z_0)^n
This series converges within a disk |z - z_0| < R_0, where R_0 is the radius of convergence. The value of R_0 is determined by the distance from z_0 to the nearest singularity of the complete analytic function. Two distinct scenarios emerge.
In the first case, the disk of convergence is entirely contained within the original domain \mathcal D_1. The power series provides a new representation of the function but does not extend its domain of definition.
In the second, the disk of convergence extends beyond the boundary of \mathcal D_1. Let the domain of convergence of the series be \mathcal D_2 = \{ z \in \mathbb{C} : |z - z_0| < R_0 \}. This new domain \mathcal D_2 has a non-empty intersection with the original domain, \mathcal D_{12} = \mathcal D_1 \cap \mathcal D_2. Inside \mathcal D_2, the power series defines an analytic function, let us call it f_2(z).
Within the common subdomain \mathcal D_{12}, both f_1(z) and f_2(z) are defined, and since f_2(z) is the Taylor series of f_1(z), they are identical there. By the uniqueness theorem, f_2(z) is the unique analytic continuation of f_1(z) into the domain \mathcal D_2.
We can then define a single analytic function over the union of the domains \mathcal D_1 \cup \mathcal D_2:
F(z) = \begin{cases} f_1(z), & \quad z \in \mathcal D_1 \\ f_2(z), & \quad z \in \mathcal D_2 \end{cases}
This process can be iterated. By selecting a new point z_1 in the newly acquired part of the domain and developing another power series, we can potentially extend the domain of analyticity even further. This generates a chain of domains \mathcal D_1, \mathcal D_2, \dots, \mathcal D_n, \dots, each new domain representing a successful analytic continuation. If this chain of domains eventually overlaps with itself in a way that yields different function values, the complete function must be defined on a Riemann surface to remain single-valued.
Example: geometric series continuation
Let us examine this procedure with a concrete example. Consider the function f_1(z) defined by the power series:
f_1(z) = \sum_{n=0}^\infty z^n
This is the geometric series, which converges for |z|<1. Its domain of definition is the open unit disk \mathcal D_1 = \{z \in \mathbb{C} : |z|<1\}, where it sums to the analytic function \frac{1}{1-z}.
Let us choose a point z_0 \in \mathcal D_1 and expand the function f_1(z) = (1-z)^{-1} into a new power series around z_0. The coefficients c_n are determined by Taylor’s theorem:
c_n = \frac{f_1^{(n)}(z_0)}{n!} = \frac{1}{n!} \left[ n!(1-z)^{-(n+1)} \right]_{z=z_0} = \frac{1}{(1-z_0)^{n+1}}
The radius of convergence R for this new series is given by the Cauchy-Hadamard formula or the ratio test. Using the latter:
R = \lim_{n\to\infty} \left| \frac{c_n}{c_{n+1}} \right| = \lim_{n\to\infty} \left| \frac{(1-z_0)^{-(n+1)}}{(1-z_0)^{-(n+2)}} \right| = |1-z_0|
If we select z_0 such that z_0 is not a non-negative real number, the new disk of convergence |z-z_0| < |1-z_0| will extend beyond the original unit disk |z|<1. The function defined by this new series is:
f_2(z) = \sum_{n=0}^\infty \frac{(z - z_0)^n}{(1-z_0)^{n+1}}
This series is the analytic continuation of f_1(z) into the domain \mathcal D_2 = \{z \in \mathbb{C} : |z - z_0| < |1 - z_0|\}. We can verify that this series also represents the function \frac{1}{1-z} within its disk of convergence by recognizing it as a geometric series:
\begin{aligned} f_2(z) &= \frac{1}{1-z_0} \sum_{n=0}^\infty \left(\frac{z - z_0}{1-z_0}\right)^n \\ &= \frac{1}{1-z_0} \left( \frac{1}{1 - \frac{z - z_0}{1-z_0}} \right) \\ &= \frac{1}{1-z_0} \left( \frac{1-z_0}{1-z_0 - (z - z_0)} \right) \\ &= \frac{1}{1-z} \end{aligned}
The sum is valid provided the common ratio has a modulus less than one, which is the condition |z-z_0| < |1-z_0|. By selecting a new point z_1 inside \mathcal D_2, we can construct a third function element, f_3(z), which converges in the disk |z-z_1| < |1-z_1|. This new function is the analytic continuation of f_2(z) and, by extension, of f_1(z).
For any choice of expansion center z_k, the boundary of the corresponding disk of convergence, |z-z_k| = |1-z_k|, always passes through the point z=1. By repeatedly applying this procedure, we can construct the analytic continuation of f_1(z) to every point in the complex plane except for z=1. The function F(z)=\frac{1}{1-z}, defined on \mathbb{C} \setminus \{1\}, is the complete analytic function obtained from the initial series element f_1(z).
Further analytic continuation beyond this domain is not possible. The point z=1 is a boundary for the domain of analyticity of F(z) and is therefore a singular point of this function.
Regular and singular points
Let f(z) be a function analytic in a domain \mathcal{D}. A point z_0 belonging to the closure of the domain, z_0 \in \overline{\mathcal{D}}, is defined as a regular point of f(z) if there exists a power series expansion centered at z_0:
\sum_{n=0}^\infty c_n(z-z_0)^n
which possesses a non-zero radius of convergence R(z_0) > 0 and coincides with the function f(z) in the intersection of its disk of convergence |z - z_0| < R(z_0) and the original domain \mathcal{D}. Any point z \in \overline{\mathcal{D}} that is not a regular point is termed a singular point.
For a function analytic in an open domain \mathcal{D}, all interior points are by definition regular points. The character of the points on the boundary \partial\mathcal{D} determines the possibilities for analytic continuation. For instance, the geometric series f(z) = \sum_{n=0}^\infty z^n converges to 1/(1-z) for |z|<1. Every point z \in \mathbb{C} \setminus \{1\} is a regular point for the complete analytic function F(z)=1/(1-z), while z=1 is its sole singular point. Similarly, for the complete analytic functions corresponding to z^{1/n} and \operatorname{Ln}(z), all points are regular except for the branch points at z=0 and z=\infty.
If an analytic function f_1(z) is initially specified in a domain \mathcal{D}_1, and all points on a connected segment of its boundary \partial\mathcal{D}^\prime \subset \partial\mathcal{D}_1 are regular, then f_1(z) can be analytically continued across \partial\mathcal{D}^\prime into a larger domain. The existence of a convergent Taylor series at each point on the boundary provides the local analytic elements needed for this continuation.
If it happens that all points on the entire boundary \partial\mathcal{D}_1 are regular, the function is said to be analytic in the closed domain \overline{\mathcal{D}_1} and can be continued into a domain that strictly contains \overline{\mathcal{D}_1}. Conversely, analytic continuation across any portion of a boundary composed exclusively of singular points is impossible. Such a boundary is known as a natural boundary.
This leads to a theorem regarding the limits of convergence for power series.
Theorem: on the boundary of the circle of convergence of a power series, there exists at least one singular point of the analytic function defined by that series.
Proof: let us consider a power series:
f(z) = \sum_{n=0}^\infty c_n(z-z_0)^n
Let its radius of convergence be R_0, with 0 < R_0 < \infty. Let K_0 denote the open disk of convergence |z-z_0| < R_0 and C_0 be its boundary circle |z-z_0| = R_0. We proceed by contradiction, assuming that every point on the circle C_0 is a regular point of the analytic function f(z).
By this assumption, for any point z^\prime \in C_0, there exists a power series representation centered at z^\prime that converges in a disk |z - z^\prime| < R(z^\prime) for some radius R(z^\prime) > 0. This series must agree with f(z) in the region where their domains overlap.
Let us establish that the function R(z^\prime) is continuous on the compact set C_0. Consider two distinct points z_1^\prime and z_2^\prime on C_0. Let the functions defined by the series at these points be f_1(z) and f_2(z), analytic in disks K_1: |z-z_1^\prime|<R(z_1^\prime) and K_2: |z-z_2^\prime|<R(z_2^\prime) respectively. Within the intersection K_0 \cap K_1 \cap K_2, both functions coincide with f(z) and therefore with each other. By the uniqueness of analytic continuation, f_1(z) and f_2(z) represent the same analytic function in K_1 \cap K_2.
Suppose that the inequality |R(z_1^\prime) - R(z_2^\prime)| \le |z_1^\prime - z_2^\prime| does not hold. For instance, assume R(z_2^\prime) > R(z_1^\prime) + |z_1^\prime - z_2^\prime|. By the triangle inequality, for any point z in the disk K_1, we have:
|z - z_2^\prime| = |(z-z_1^\prime) + (z_1^\prime - z_2^\prime)| \le |z-z_1^\prime| + |z_1^\prime - z_2^\prime| < R(z_1^\prime) + |z_1^\prime - z_2^\prime| < R(z_2^\prime)
This shows that the entire disk K_1 is contained within the disk K_2.
Since f_1(z) and f_2(z) are the same function where both are defined, and K_1 \subset K_2, the function f_1(z) (originally defined by a series with radius of convergence R(z_1^\prime)) is analytic throughout the larger disk K_2. This implies that the radius of convergence of the series for f_1(z) must be at least R(z_2^\prime). This contradicts the assumption that R(z_2^\prime) > R(z_1^\prime) + |z_1^\prime - z_2^\prime|. The contradiction forces to accept the inequality:
|R(z_1^\prime) - R(z_2^\prime)| \le |z_1^\prime - z_2^\prime|
This condition implies that R(z^\prime) is a Lipschitz continuous function on the circle C_0. Since C_0 is a compact set, the continuous function R(z^\prime) must attain its minimum value on C_0. Let this minimum be R_{min} = \min_{z^\prime \in C_0} R(z^\prime). As we assumed R(z^\prime) > 0 for all z^\prime \in C_0, it follows that R_{min} > 0.
The collection of open disks |z-z^\prime| < R(z^\prime) for every z^\prime \in C_0 forms an open cover of the compact set C_0. A finite number of these disks is sufficient to cover C_0. These disks, together with the original disk K_0, define a single-valued analytic function F(z) that is analytic in the union of all these disks. This union contains the larger open disk |z-z_0| < R_0 + R_{min}.
If F(z) is analytic in |z-z_0| < R_0 + R_{min}, then its Taylor series expansion around z_0 must converge throughout this disk. But F(z) coincides with f(z) inside K_0, so its Taylor series at z_0 is the original series \sum c_n(z-z_0)^n. This leads to the conclusion that the radius of convergence of the original series must be at least R_0 + R_{min}. This contradicts the initial hypothesis that the radius of convergence is R_0.
The contradiction was derived from the assumption that all points on the circle of convergence are regular. Therefore, this assumption must be false. There must be at least one singular point on the boundary C_0.
From this theorem, it follows that the radius of convergence of a power series is determined by the distance from the center of expansion to the nearest singular point of the complete analytic function which the series represents.
Complete analytic function
The mechanism of analytic continuation permits the extension of a function, initially defined as an analytic element (f_1, \mathcal{D}_1), into a more extensive domain. This extension is constructed by forming a chain of analytic elements (f_1, \mathcal{D}_1), (f_2, \mathcal{D}_2), \dots, (f_n, \mathcal{D}_n), where for each i \in \{1, \dots, n-1\}, the intersection \mathcal{D}_i \cap \mathcal{D}_{i+1} is a non-empty open set, and the functions agree on this intersection, f_i(z) = f_{i+1}(z) for all z \in \mathcal{D}_i \cap \mathcal{D}_{i+1}.
This process results in a single-valued analytic function defined on the union of the domains \mathcal{D} = \bigcup_{i=1}^n \mathcal{D}_i or, in the case the function is multi-valued, on a corresponding Riemann surface. By considering not just one chain, but all possible chains of analytic continuation emanating from the initial element (f_1, \mathcal{D}_1), we arrive at the most comprehensive representation of the function.
The complete analytic function is defined as the collection of all analytic elements that can be generated from an initial element (f_1, \mathcal{D}_1) through all possible paths of analytic continuation. The domain of definition for this complete function is called its natural domain of existence. This domain is the union of all individual domains associated with every analytic element in the collection.
The natural domain of existence, which we can denote by \mathcal{R}, is not in general a simple domain within the complex plane. Frequently, \mathcal{R} is a Riemann surface, upon which the complete analytic function F(z) is single-valued and analytic. A property of this construction is that the complete analytic function cannot be extended beyond its natural domain. The boundary of this domain acts as a terminal barrier for the process of analytic continuation.
This implies that all points on the boundary of the natural domain of existence must be singular points for the complete analytic function. We can demonstrate this by contradiction.
Let F(z) be a complete analytic function with its natural domain of existence \mathcal{R}. Assume there exists a point z_0 on the boundary \partial\mathcal{R} that is a regular point of F(z). According to the definition of a regular point, there must exist a function G(z) that is analytic in an open disk |z - z_0| < R(z_0) for some R(z_0)>0. This function G(z) must coincide with F(z) on the intersection of this disk with the domain \mathcal{R}.
Since z_0 is a boundary point of \mathcal{R}, the disk |z - z_0| < R(z_0) must contain points that are inside \mathcal{R} and points that are outside \mathcal{R}. The pair (G(z), \{z: |z - z_0| < R(z_0)\}) constitutes an analytic element. This element provides an analytic continuation of F(z) into a region beyond its supposed natural domain of existence.
This result contradicts the definition of the complete analytic function and its natural domain, which by construction already incorporates all possible analytic continuations. The initial assumption that a regular point can exist on the boundary must therefore be false. Consequently, every point on the boundary of the natural domain of existence of a complete analytic function is a singular point. Such a boundary is referred to as a natural boundary.