# Analytic continuation

of a function

An extension of a function , already defined on a certain subset of a complex manifold , to a function which is holomorphic on a certain domain containing such that the restriction of to coincides with . The starting point in the theory of analytic continuation is the concept of an (analytic) element, i.e. a pair , where is a domain in and is a holomorphic function on . One says that the elements and are direct analytic continuations of each other through a connected component of the set if . By definition, an element continues analytically to a boundary point if there exists a direct analytic continuation of the element through such that . A maximal analytic continuation of (in ) is an element which continues analytically to the domain , but which cannot analytically be continued to any boundary point of . The maximal analytic continuation of in is unique, but does not always exist. In order to overcome this drawback one introduces the concept of a covering domain over (a Riemann surface in the case ), which is constructed from the elements that are analytic continuations of . An element is called an analytic continuation of an element if there exists a finite chain of elements , , and corresponding connected components in such that and , are direct analytic continuations of each other through . One says that the holomorphic function , initially defined in the domain , continues analytically to a point if there exists an analytic continuation of such that . One introduces an equivalence relation among the elements which are continuations of to the point if and in a neighbourhood of . On the set of equivalence classes (for all possible ) there is a natural way of introducing the topology and complex structure of a covering domain over . The function is lifted to in a natural manner (its value on the equivalence class at containing is set equal to ); it continues analytically to all of and, in the sense defined above, it does not continue to any boundary point of over .

If is the complex plane or, more generally, the complex space , , this process of analytic continuation can be described more simply. A canonical element is a pair where and is a power series centred at the point with a non-empty domain of convergence . A canonical element is an analytic continuation of along a path if there exists a family of canonical elements , , with centres , such that , , and for any the elements are direct analytic continuations of for all sufficiently near to . The family is in fact uniquely determined. If , , is a continuous family of paths in with common end points and , and if continues analytically along each , then the result does not depend on (the monodromy theorem). In the case of the canonical elements obtained by analytic continuation along all possible paths in become the points of ; is lifted to and continues analytically throughout to a holomorphic function , while is the domain of holomorphy of .

This general process of analytic continuation is not very effective in practice, and for this reason many special methods of analytic continuation are employed. These include various analytic representations: parameter-dependent integrals such as Cauchy-type integrals (cf. Cauchy integral); the Laplace integral; the Borel transform; change of variable in a power series, special methods of summation of power series (the Borel expansion into a polynomial series converging in a maximal polygon (cf. Borel summation method), the Mittag-Leffler series converging in a maximal star (cf. Star of a function element; Mittag-Leffler summation method), etc., the Riemann–Schwarz reflection method (cf. Riemann–Schwarz principle), functional and differential equations satisfied by a function (e.g. the equation for the gamma-function, conditions of periodicity, evenness, symmetry, etc.), and analytic expressions in terms of known functions.

The subject of analytic continuation also comprises studies on the relation between the initial element of an analytic function (a Taylor series) and the properties of the complete analytic function generated by this element [1]. Results have been obtained on singular points (criteria for singular points, the Hadamard theorem on products, the Fabry theorem on quotients) and on singular curves (theorems on gaps and non-extendability beyond the boundary of the disc of convergence, e.g. the Hadamard theorem on gaps, the Fabry theorem on gaps, etc.), theorems on over-convergence and on relations between analytic continuation of a power series and properties of an entire function which defines its coefficients, problems of meromorphic continuation, meromorphic continuation by Padé approximation, etc. The field of analytic continuation also includes theorems on removable singularities (removability of an isolated singularity of a bounded holomorphic function, removability of a rectifiable singular curve under conditions of continuity, etc.), as well as a large class of theorems on the simultaneous continuation of holomorphic functions of several complex variables. The space , , comprises domains for which every holomorphic function can be extended to a larger domain (this phenomenon is absent in the one-dimensional case). It is therefore an important problem in the theory of analytic continuation of functions of several complex variables to describe these larger domains — the so-called envelopes of holomorphy. Thus, there are descriptions of the hulls (envelopes) of holomorphy for Hartogs domains, -circular and tube domains, theorems on the removability of compact singularity sets and of singularity sets of codimension , the Bogolyubov theorem on the "edge-of-the-wedge" and

P Vladimirov's theorem on -convex hulls [3] (cf. Holomorphic envelope; Hartogs domain; Tube domain). A number of effective methods for the construction of envelopes of holomorphy are available [3].

The problem of analytic continuation of functions of a real variable can be reduced to that of holomorphic functions, since for any domain and any function analytic in there are a domain and a function holomorphic in such that and .

#### References

 [1] L. Bieberbach, "Analytische Fortsetzung" , Springer (1955) pp. Sect. 3 MR0068621 Zbl 0064.06902 [2] A.I. Markushevich, "Theory of functions of a complex variable" , 1–3 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002 [3] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)