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 the infinite series representation that initially defined the function becomes divergent.

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 singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology.

Initial discussion

Suppose f is an analytic function defined on a non-empty open subset U of the complex plane If V is a larger open subset of containing U, and F is an analytic function defined on V such that

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 the connected domain of two analytic functions F1 and F2 such that U is contained in V and for all z in U

then

on all of V. This is because F1−F2 is an analytic function which vanishes on the open, connected domain U of f and hence must vanish on its entire domain. This follows directly from the identity theorem for holomorphic functions.

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.

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.

Analytic continuation is used in Riemannian manifolds, in the context of solutions of Einstein's equations. For example, Schwarzschild coordinates can be analytically continued into Kruskal–Szekeres coordinates.

Worked example

. In this case, it is given by a power series:

By the Cauchy–Hadamard theorem.

(because it is a geometric series:

.

. Using Cauchy's differentiation formula

The last summation results from the th derivation of the geometric series

are the same when the successive centers have a positive imaginary part or a negative imaginary part. This is not always the case; in particular this is not the case for the complex logarithm, the antiderivative of the above function.

Formal definition of a germ

The power series defined below is generalized by the idea of a germ. The general theory of analytic continuation and its generalizations is known as sheaf theory. Let

be a power series converging in the disk Dr(z0), r > 0, defined by

1. .

Note that without loss of generality, here and below, we will always assume that a maximal such r was chosen, even if that r 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

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.

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.

The topology of the set of germs

Let g and h be germs where r is the radius of convergence of g and if the power series defined by g and h specify 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.

We can define a topology. Let r > 0, and let

The sets Ur(g), for all r define a basis of open sets.

A connected component (i.e., an equivalence class) is called a sheaf where r is the radius of convergence of g, is a chart. The set of such charts forms an atlas is a Riemann surface is sometimes called the universal analytic function.

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

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.

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.

Natural boundary

Suppose that a power series has radius of convergence r and defines an analytic function f inside that disc. Consider points on the circle of convergence. A point for which there is a neighbourhood on which f has an analytic extension is regular, otherwise singular. The circle is a natural boundary if all its points are singular.

More generally, we may apply the definition to any open connected domain on which f is analytic, and classify the points of the boundary of the domain as regular or singular: the domain boundary is then a natural boundary if all points are singular, in which case the domain is a domain of holomorphy.

Example I: A function with a natural boundary at zero (the prime zeta function)

we define the so-called prime zeta function, to be

This function is analogous to the summatory form of the Riemann zeta function, except with indices restricted only to the prime numbers instead of taking the sum over all positive natural numbers. The prime zeta function has an analytic continuation to all complex s by the logarithms of the Riemann zeta function as

. Since the set of points

has accumulation point has no analytic continuation for s in an essential way.

Example II: A typical lacunary series (natural boundary as subsets of the unit circle)

, we define the lacunary series of order c by the power series expansion

for any z given by

For any positive natural numbers c to other complex z-th roots of unity. Hence, since the set formed by all such roots is dense on the boundary of the unit circle to complex z whose modulus exceeds one.

, let

denotes the open unit disk distinct complex numbers z to show that

Thus for any arc on the boundary of the unit circle, there are an infinite number of points z Hence, there is no analytic continuation for these functions beyond the interior of the unit circle.

Monodromy theorem

The monodromy theorem gives a sufficient condition for the existence of a direct analytic continuation (i.e., an extension of an analytic function to an analytic function on a bigger set).

is an open set and f an analytic function on D. If G is a simply connected domain containing D, such that f has an analytic continuation along every path in G, starting from some fixed point a in D, then f has a direct analytic continuation to G.

In the above language this means that if G is a simply connected domain, and S is a sheaf whose set of base points contains G, then there exists an analytic function f on G whose germs belong to S.

Hadamard's gap theorem

For a power series

with

the circle of convergence is a natural boundary. Such a power series is called lacunary. This theorem has been substantially generalized by Eugène Fabry (see Fabry's gap theorem) and George Pólya.

Pólya's theorem

Let

be a power series. Then there exist εk ∈ {−1, 1} such that

has the convergence disc of f around z0 as a natural boundary.

The proof of this theorem makes use of Hadamard's gap theorem.

See also

• Mittag-Leffler star

• Holomorphic functional calculus

• Numerical analytic continuation

• Polya's shire theorem

References

External links

• at MathPages