Analytic continuation


In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain 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 becomes divergent.
The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies. 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.

Worked example

Begin with a particular analytic function. In this case, it is given by a power series centered at :
By the Cauchy–Hadamard theorem, its radius of convergence is 1. That is, is defined and analytic on the open set which has boundary. Indeed, the series diverges at.
Pretend we don't know that, and focus on recentering the power series at a different point :
We'll calculate the 's and determine whether this new power series converges in an open set which is not contained in. If so, we will have analytically continued to the region which is strictly larger than.
The distance from to is. Take ; let be the disk of radius around ; and let be its boundary.
Then. Using Cauchy's differentiation formula to calculate the new coefficients,
That is,
which has radius of convergence, and If we choose with, then is not a subset of and is actually larger in area than. The plot shows the result for
We can continue the process: select, recenter the power series at, and determine where the new power series converges. If the region contains points not in, then we will have analytically continued even farther. This particular can be analytically continued to the punctured complex plane

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, r > 0, defined by
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 and the top g1 of g is α0. The top of g is the value of f at z0.
Any vector g = 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 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 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. This extension by transitivity is one definition of analytic continuation. The equivalence relation will be denoted.
We can define a topology on. Let r > 0, and let
The sets Ur, for all r > 0 and define a basis of open sets for the topology on.
A connected component of is called a sheaf. We also note that the map defined by 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.

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 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 then this function will have the property that exp = 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)

For we define the so-called prime zeta function,, to be
This function is analogous to the summatory form of the Riemann zeta function when in so much as it is the same summatory function as, 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 such that, a fact which follows from the expression of by the logarithms of the Riemann zeta function as
Since has a simple, non-removable pole at, it can then be seen that has a simple pole at. Since the set of points
becomes dense in as we approach zero, or equivalently,
we can see that zero forms a natural boundary for. This implies that has no analytic continuation for s left of zero, i.e., there is no continuation possible for when. As a remark, this fact can be problematic if we are performing a complex contour integral over an interval whose real parts are symmetric about zero, say for some, where the integrand is a function with denominator that depends on in an essential way.

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

For integers, we define the lacunary series of order c by the power series expansion
Clearly, since there is a functional equation for for any z satisfying given by. It is also not difficult to see that for any integer, we have another functional equation for given by
For any positive natural numbers c, the lacunary series function has a simple pole at. We consider the question of analytic continuation of to other complex z such that As we shall see, for any, the set of -th roots of unity impose a natural boundary on the function. Hence, since the set of the union of all such roots of unity over is dense on the boundary of the unit circle, we have no possible analytic continuation of to complex z whose real parts exceed one.
The proof of this fact is generalized from a standard argument for the case where Namely, for integers, let
where denotes the open unit disk in the complex plane and, i.e., there are distinct complex numbers z that lie on or inside the unit circle such that. Now the key part of the proof is to use the functional equation for when to show that
Thus for any arc on the boundary of the unit circle, there are an infinite number of points z within this arc such that. This condition is equivalent to saying that the circle forms a natural boundary for the function for any fixed choice of 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.
Suppose 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 Eugen Fabry and George Pólya.

Pólya's theorem

Let
be a power series, then there exist εk ∈ 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.

A useful theorem: A sufficient condition for analytic continuation to the non-positive integers

In most cases, if an analytic continuation of a complex function exists, it is given by an integral formula. The next theorem, provided its hypotheses are met, provides a sufficient condition under which we can continue an analytic function from its convergent points along the positive reals to arbitrary . Moreover, the formula gives an explicit representation for the values of the continuation to the non-positive integers expressed exactly by higher order derivatives of the original function evaluated at zero.

Hypotheses of the theorem

We require that a function satisfies the following conditions in order to apply the theorem on continuation of this function stated below:
Let F be any function defined on the positive reals that satisfies all of the conditions - above. Then the integral representation of the scaled Mellin transform of F at s, denoted by, has an meromorphic continuation to the complex plane. Moreover, we have that for any non-negative, the continuation of F at the point is given explicitly by the formula

Examples

Example I: The connection of the Riemann zeta function to the Bernoulli numbers

We can apply the theorem to the function
which corresponds to the exponential generating function of the Bernoulli numbers,. For, we can express, since we can compute that the next integral formula for the reciprocal powers of the integers holds for s in this range:
Now since the integrand of the last equation is a uniformly continuous function of t for each positive integer n, we have an integral representation for whenever given by
When we perform integration by parts to the Mellin transform integral for this, we also obtain the relation that
Moreover, since for any fixed integer polynomial power of t, we meet the hypothesis of the theorem which requires that. The standard application of Taylor's theorem to the ordinary generating function of the Bernoulli numbers shows that. In particular, by the observation made above to shift, and these remarks, we can compute the values of the so-called trivial zeros of the Riemann zeta function and the rational-valued negative odd integer order constants,, according to the formula

Example II: An interpretation of ''F'' as the summatory function for some arithmetic sequence

Suppose that F is a smooth, sufficiently decreasing function on the positive reals satisfying the additional condition that
In application to number theoretic contexts, we consider such F to be the summatory function of the arithmetic function f,
where we take and the prime-notation on the previous sum corresponds to the standard conventions used to state Perron's theorem:
We are interested in the analytic continuation of the DGF of f, or equivalently of the Dirichlet series over f at s,
Typically, we have a particular value of the abscissa of convergence,, defined such that is absolutely convergent for all complex s satisfying, and where is assumed to have a pole at and so that the initial Dirichlet series for diverges for all s such that. It is known that there is a relationship between the Mellin transform of the summatory function of any f to the continuation of its DGF at of the form:
That is to say that, provided has a continuation to the complex plane left of the origin, we can express the summatory function of any f by the inverse Mellin transform of the DGF of f continued to s with real parts less than zero as:
We can form the DGF, or Dirichlet generating function, of any prescribed f given our smooth target function F by performing summation by parts as
where is the Laplace-Borel transform of F, which if
corresponds to the exponential generating function of some sequence enumerated by , then
is its ordinary generating function form over the sequence whose coefficients are enumerated by.
So it follows that if we write
alternately interpreted as a signed variant of the binomial transform of F, then we can express the DGF as the following Mellin transform at :
Finally, since the gamma function has a meromorphic continuation to, for all we have an analytic continuation of the DGF for f at -s of the form
where a formula for for non-negative integers n is given according to the formula in the theorem as
Moreover, provided that the arithmetic function f satisfies so that its Dirichlet inverse function exists, the DGF of is continued to any, that is any complex s excluding s in a f-defined, or application dependent f-specific, so-called critical strip between the vertical lines, and the value of this inverse function DGF when is given by
To continue the DGF of the Dirichlet inverse function to s inside this f-defined critical strip, we must require some knowledge of a functional equation for the DGF,, that allows us to relate the s such that the Dirichlet series that defines this function initially is absolutely convergent to the values of s inside this strip -- in essence, a formula providing that is necessary to define the DGF in this strip.