Analytic continuation

of a function

An extension of a function $ f _ {0} $, already defined on a certain subset $ E $ of a complex manifold $ M $, to a function $ f $ which is holomorphic on a certain domain $ D \subset M $ containing $ E $ such that the restriction $ f \mid_ {E} = f _ {0} $ of $ f $ to $ E $ coincides with $ f _ {0} $. The starting point in the theory of analytic continuation is the concept of an (analytic) element, i.e. a pair $ (D , f) $, where $ D $ is a domain in $ M $ and $ f $ is a holomorphic function on $ D $. One says that the elements $ (D _ {0} , f _ {0} ) $ and $ ( D _ {1} , f _ {1} ) $ are direct analytic continuations of each other through a connected component $ \Delta $ of the set $ D _ {0} \cap D _ {1} $ if $ f _ {0} | _ \Delta = f _ {1} | _ \Delta $. By definition, an element $ (D _ {0} , f _ {0} ) $ continues analytically to a boundary point $ \xi \in \partial D \subset M $ if there exists a direct analytic continuation $ (D _ {1} , f _ {1} ) $ of the element $ ( D _ {0} , f _ {0} ) $ through $ \Delta $ such that $ \xi \in {\overline \Delta } \cap D _ {1} $. A maximal analytic continuation of $ ( D _ {0} , f _ {0} ) $ (in $ M $) is an element $ ( D, f) $ which continues $ f _ {0} $ analytically to the domain $ D \supset D _ {0} $, but which cannot analytically be continued to any boundary point of $ D $. The maximal analytic continuation of $ ( D _ {0} , f _ {0} ) $ in $ M $ is unique, but does not always exist. In order to overcome this drawback one introduces the concept of a covering domain over $ M $ (a Riemann surface in the case $ M = \mathbf C $), which is constructed from the elements that are analytic continuations of $ ( D _ {0} , f _ {0} ) $. An element $ (D, f) $ is called an analytic continuation of an element $ ( D _ {0} , f _ {0} ) $ if there exists a finite chain of elements $ ( D _ {i} , f _ {i} ) $, $ i = 0 \dots n $, and corresponding connected components $ \Delta _ {i} $ in $ D _ {i} \cap D _ {i+1 } $ such that $ ( D _ {n} , f _ {n} ) = ( D , f ) $ and $ ( D _ {i} , f _ {i} ) $, $ ( D _ {i+1 } , f _ {i+1 } ) $ are direct analytic continuations of each other through $ \Delta _ {i} $. One says that the holomorphic function $ f _ {0} $, initially defined in the domain $ D _ {0} $, continues analytically to a point $ z \in M $ if there exists an analytic continuation $ (D, f) $ of $ ( D _ {0} , f _ {0} ) $ such that $ z \in D $. One introduces an equivalence relation among the elements which are continuations of $ f _ {0} $ to the point $ z: $ $ ( D ^ \prime , f ^ { \prime } ) \sim ( D ^ {\prime\prime} , f ^ { \prime\prime } ) $ if $ z \in D ^ \prime \cap D ^ {\prime\prime} $ and $ f ^ { \prime } = f ^ { \prime\prime } $ in a neighbourhood of $ z $. On the set of equivalence classes $ D _ {f} $ (for all possible $ z $) there is a natural way of introducing the topology and complex structure of a covering domain over $ M $. The function $ f _ {0} $ is lifted to $ D _ {f} $ in a natural manner (its value on the equivalence class at $ z $ containing $ (D _ {0} , f _ {0} ) $ is set equal to $ f _ {0} (z) $); it continues analytically to all of $ D _ {f} $ and, in the sense defined above, it does not continue to any boundary point of $ D _ {f} $ over $ M $.

If $ M $ is the complex plane $ \mathbf C $ or, more generally, the complex space $ \mathbf C ^ {n} $, $ n \geq 1 $, this process of analytic continuation can be described more simply. A canonical element is a pair $ ( D _ {a} , f _ {a} ) $ where $ a \in \mathbf C ^ {n} $ and $ f _ {a} $ is a power series centred at the point $ a $ with a non-empty domain of convergence $ D _ {a} $. A canonical element $ (D _ {b} , f _ {b} ) $ is an analytic continuation of $ ( D _ {a} , f _ {a} ) $ along a path $ \gamma : [ 0, 1 ] \rightarrow \mathbf C ^ {n} $ if there exists a family of canonical elements $ ( D _ {t} , f _ {t} ) $, $ 0 \leq t \leq 1 $, with centres $ a _ {t} = z(t) $, such that $ ( D _ {0} , f _ {0} ) = ( D _ {a} , f _ {a} ) $, $ ( D _ {1} , f _ {1} ) = ( D _ {b} , f _ {b} ) $, and for any $ t _ {0} \in \gamma $ the elements $ ( D _ {t} , f _ {t} ) $ are direct analytic continuations of $ ( D _ {t _ {0} } , f _ {t _ {0} } ) $ for all $ t $ sufficiently near to $ t _ {0} $. The family $ ( D _ {t} , f _ {t} ) $ is in fact uniquely determined. If $ \gamma _ \tau $, $ 0 \leq \tau \leq 1 $, is a continuous family of paths in $ \mathbf C ^ {n} $ with common end points $ a $ and $ b $, and if $ ( D _ {a} , f _ {a} ) $ continues analytically along each $ \gamma _ \tau $, then the result $ ( D _ {b} , f _ {b} ) $ does not depend on $ \tau $ (the monodromy theorem). In the case of $ \mathbf C ^ {n} $ the canonical elements $ ( D _ {a} , f _ {a} ) $ obtained by analytic continuation along all possible paths in $ \mathbf C ^ {n} $ become the points of $ D _ {f} $; $ f _ {a} $ is lifted to $ D _ {f} $ and continues analytically throughout $ D _ {f} $ to a holomorphic function $ f $, while $ D _ {f} $ is the domain of holomorphy of $ f $.

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 $ \Gamma (z + 1) = z \Gamma (z) $ 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 $ \mathbf C ^ {n} $, $ n > 1 $, 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, $ n $-circular and tube domains, theorems on the removability of compact singularity sets and of singularity sets of codimension $ \geq 2 $, the Bogolyubov theorem on the "edge-of-the-wedge" and Vladimirov's theorem on $ C $-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 $ G \subset \mathbf R ^ {n} $ and any function $ f $ analytic in $ G $ there are a domain $ D \subset \mathbf C ^ {n} $ and a function $ \widetilde{f} $ holomorphic in $ D $ such that $ D \cap \mathbf R ^ {n} = G $ and $ \widetilde{f} \mid _ {G} = f $.

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)
[a1]	L. Hörmander, "An introduction to complex analysis in several variables" , v. Nostrand (1966) MR0203075 Zbl 0138.06203
[a2]	H. Grauert, K. Fritzsche, "Several complex variables" , Springer (1976) (Translated from German) MR0414912 Zbl 0381.32001