In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined makes no good sense. Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In mathematics, a series is a sum of a sequence of terms. ...

The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of mathematical singularities. The case of several complex variables is rather different, since singularities then cannot be isolated points, and its investigation was a major reason for the development of sheaf cohomology. In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ... The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ... In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric...

Initial discussion

Suppose f is an analytic function defined on an open subset U of the complex plane C. If V is a larger open subset of C, containing U, and F is an analytic function defined on V such that In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

F(z) = f(z) for all z in U,

then F is called an analytic continuation of f. In other words, the restriction of F to U is the function f we started with.

Analytic continuations are unique in the following sense: if V is connected and F₁ and F₂ are two analytic continuations of f defined on V, then In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...

F₁ = F₂

everywhere. That is because the difference is an analytic function which vanishes on the intersection of their domains, a non-empty open set, and an analytic function which vanishes on a non-empty open set must vanish everywhere on its domain (assuming the domain is connected) and hence must be identically zero. Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ...

For example, if a power series with radius of convergence r about a point a of C is given, one can consider analytic continuations of the power series, i.e. analytic functions F which are defined on larger sets than the open disc of radius r at a, in symbols In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... A disk is the inside of a circle. ...

{z : |z − a| < r},

and agree with the given power series on that set. The number r is maximal in the following sense: there always exists a complex number z with

|z − a| = r

such that no analytic continuation of the series can be defined at z. Therefore there is a limitation to analytic continuation to bigger discs with the same centre a. On the other hand there may well be analytic continuations to some larger sets. That depends on the radius of convergence when you expand about points b other than a; if that is greater than

r − |b − a|

then we win the right to use that expansion on an open disc, part of which lies outside the original disc of definition. If not, there is a natural boundary on the bounding circle.

Applications

A common way to define functions in complex analysis proceeds by first specifying the function on a small domain only, and then extending it by analytic continuation. In practice, this continuation is often done by first establishing some functional equation on the small domain and then using this equation to extend the domain. Examples are the Riemann zeta function and the gamma function. In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ... In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...

The concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function. The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea of Riemann surfaces. In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...

The power series defined above is generalized by the idea of a germ. The general theory of analytic continuation and its generalization is known as sheaf theory. In mathematics, a germ is an equivalence class of continuous functions from one topological space to another (often from the real line to itself), in which one point x0 in the domain has been singled out as privileged. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...

Formal definition of a germ

Let

be a power series converging in the disc D_r(z₀) := {z in C : |z - z₀| < r} for r > 0. (Note, without loss of generality, here and in the sequel, we will always assume that a maximal such r was chosen, even if it is ∞.) Also note that it would be equivalent to begin with an analytic function defined on some small open set. We say that the vector In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

g = (z₀, α₀, α₁, α₂, ...)

is a germ of f. The base g₀ of g is z₀, the stem of g is (α₀, α₁, α₂, ...) and the top g₁ of g is α₀. The top of g is the value of f at z₀, the bottom of g. In mathematics, a germ is an equivalence class of continuous functions from one topological space to another (often from the real line to itself), in which one point x0 in the domain has been singled out as privileged. ...

Any vector g = (z₀, α₀, α₁, ...) is a germ if it represents a power series of an analytic function around z₀ with some radius of convergence r > 0. Therefore, we can safely speak of the set of germs .

The topology of the set of germs

Let g and h be germs. If |h₀ - g₀| < r where r is the radius of convergence of g and if the power series that g and h represent define identical functions on the intersection of the two domains, then we say that h is generated by (or compatible with) g, and we write g ≥ h. This compatibility condition is neither transitive, symmetric nor antisymmetric. If we extend the relation by transitivity, we obtain a symmetric relation, which is therefore also an equivalence relation on germs (but not an ordering). This extension by transitivity is one definition of analytic continuation. The equivalence relation will be denoted . In mathematics, a germ is an equivalence class of continuous functions from one topological space to another (often from the real line to itself), in which one point x0 in the domain has been singled out as privileged. ... In mathematics, the transitive closure of a binary relation R on a set X is the smallest transitive relation on X that contains R. For any relation R the transitive closure of R always exists. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...

We can define a topology on . Let r > 0, and let Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...

The sets U_r(g), for all r > 0 and g ∈ define a basis of open sets for the topology on . In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases...

A connected component of (i.e., an equivalence class) is called a sheaf. We also note that the map φ_g(h) = h₀ from U_r(g) to C where r is the radius of convergence of g, is a chart. The set of such charts forms an atlas for , hence is a Riemann surface. is sometimes called the universal analytic function. Connected components come up in topology and in graph theory, two related branches of mathematics. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...

Examples of analytic continuation

is a power series corresponding to the natural logarithm near z = 1. This power series can be turned into a germ The natural logarithm is the logarithm to the base e, where e is equal to 2. ... In mathematics, a germ is an equivalence class of continuous functions from one topological space to another (often from the real line to itself), in which one point x0 in the domain has been singled out as privileged. ...

g = (1, 0, 1, −1, 1, −1, 1, −1, ...)

This germ has a radius of convergence of 1, and so there is a sheaf S corresponding to it. This is the sheaf of the logarithm function. In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...

The uniqueness theorem for analytic functions also extends to sheaves of analytic functions: if the sheaf of an analytic function contains the zero germ (i.e., the sheaf is uniformly zero in some neighborhood) then the entire sheaf is zero. Armed with this result, we can see that if we take any germ g of the sheaf S of the logarithm function, as described above, and turn it into a power series f(z) then this function will have the property that exp(f(z))=z. If we had decided to use a version of the inverse function theorem for analytic functions, we could construct a wide variety of inverses for the exponential map, but we would discover that they are all represented by some germ in S. In that sense, S is the "one true inverse" of the exponential map.

In older literature, sheaves of analytic functions were called multi-valued functions. See sheaf for the general concept. This diagram does not represent a true function; because the element 3, in X, is associated with two elements b and c, in Y. In mathematics, a multivalued function is a total relation; i. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...

Hadamard's gap theorem

For a power series

with coefficients mostly zero in the precise sense that they vanish outside a sequence of exponents k(i) with

k(i + 1)/k(i) > 1 + δ

for some fixed δ > 0, the circle centre z₀ and with radius the radius of convergence is a natural boundary. (See for example E. C. Titchmarsh, Theory of Functions.) Edward Charles (Ted) Titchmarsh (born 1 June 1899 in Newbury died 18 January 1963 at Oxford) was a leading British mathematician. ...