Analytic continuation


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 which 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

Analytic continuation of natural logarithm (imaginary part)

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




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.



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, solutions of Einstein's equations. For example, the analytic continuation of Schwarzschild coordinates into Kruskal–Szekeres coordinates.[1]

Worked example

Analytic continuation from U (centered at 1) to V (centered at a=(3+i)/2)

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, one has  

The last summation results from the kth derivation of the geometric series, which gives the formula  


which has radius of convergence   around  . 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 further. This particular   can be analytically continued to the whole punctured complex plane  

In this particular case the obtained values of   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


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 > 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 (or compatible with) g, and we write gh. 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  .

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


The sets Ur(g), for all r > 0 and   define a basis of open sets for the topology on  .

A connected component of   (i.e., an equivalence class) 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 (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)


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


has accumulation point 0 (the limit of the sequence as  ), we can see that zero forms a natural boundary for  . This implies that   has no analytic continuation for s left of (or at) 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 diverges at  . We consider the question of analytic continuation of   to other complex z such that   As we shall see, for any  , the function   diverges at the  -th roots of unity. Hence, since the set formed by all such roots is dense on the boundary of the unit circle, there is no analytic continuation of   to complex z whose modulus exceeds one.

The proof of this fact is generalized from a standard argument for the case where  [2] 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 (i.e., an extension of an analytic function to an analytic function on a bigger set).

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




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

Pólya's theorem




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.

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   (with the exception of at finitely-many poles). Moreover, the formula gives an explicit representation for the values of the continuation to the non-positive integers expressed exactly by higher order (integer) derivatives of the original function evaluated at zero.[3]

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:

  • (T-1). The function must have continuous derivatives of all orders, i.e.,  . In other words, for any integers  , the integral-order   derivative   must exist, be continuous on  , and itself be differentiable, so that all higher order derivatives of F are smooth functions of x on the positive real numbers;
  • (T-2). We require that the function F is rapidly decreasing in that for all   we obtain the limiting behavior that   as t becomes unbounded, tending to infinity;
  • (T-3). The (reciprocal gamma-scaled) Mellin transform of F exists for all complex s such that   with the exception of   (or for all s with positive real parts except possibly at a finite number of exceptional poles):

The conclusion of the theorem


Let F be any function defined on the positive reals that satisfies all of the conditions (T1)-(T3) 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




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 (for  ) 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:[4]


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   (as prescribed by the Taylor series expansion of F about zero), 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 [5]


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.[6]

See also



  1. ^ Kruskal, M. D. (1960-09-01). "Maximal Extension of Schwarzschild Metric". Physical Review. 119 (5): 1743–1745. Bibcode:1960PhRv..119.1743K. doi:10.1103/PhysRev.119.1743.
  2. ^ See the example given on the MathWorld page for natural boundary.
  3. ^ See the article Fontaine's rings and p-adic L-functions by Pierre Colmez found at this link (Course notes PDF dated 2004).
  4. ^ Much more, in fact, can be said about the properties of such relations between the continuations of a DGF and the summatory function of any arithmetic f -- and, for a short list and compendia of identities, see the working sandbox page at Dirichlet series inversion. Some interesting pairs of the summatory-function-to-DGF inversion relations that arise in non-standard applications include:  , where   is the Mertens function, or summatory function of the Moebius function,   is the prime zeta function, and   is the Riemann prime-counting function.
  5. ^ One observation on how to reconcile how the values of this analytically continued DGF coincide with what we know of the Mellin integral of the summatory function of f, we observe that we should have that
  6. ^ This construction is noted to be similar to the known functional equation for the Riemann zeta function which relates   for   to the values of   for   in the classical critical strip where we can find all of the non-trivial zeros of this zeta function.
  • Lars Ahlfors (1979). Complex Analysis (3 ed.). McGraw-Hill. pp. 172, 284.
  • Ludwig Bieberbach (1955). Analytische Fortsetzung. Springer-Verlag.
  • P. Dienes (1957). The Taylor series: an introduction to the theory of functions of a complex variable. New York: Dover Publications, Inc.