Uniqueness properties of analytic functions
Properties of analytic functions that say that they are completely determined by their values on certain subsets of their domain of definition, or of its boundary; in this connection one may distinguish interior uniqueness properties and boundary uniqueness properties.
Interior uniqueness properties.
Let be a domain in the complex plane . The classical interior uniqueness theorem for holomorphic (that is, single-valued analytic) functions on states that if two holomorphic functions and in coincide on some set containing at least one limit point in , then everywhere in . In other words, if a holomorphic function in vanishes on a set having at least one limit point in , then . The proof of this interior uniqueness property of analytic functions shows that it is essentially a uniqueness property of power series in one complex variable . The uniqueness property remains valid for meromorphic functions and in if one regards the poles of and as points at which the functions take the value .
In particular, if two analytic functions and coincide in an arbitrarily small neighbourhood of some point or on an arbitrarily small arc of some continuous curve, then . Another corollary: The set of -points of an analytic function , that is, the set of points at which , cannot have limit points inside the domain of definition , provided that .
Two complete analytic functions in the sense of Weierstrass (cf. Complete analytic function) , are, in general, multiple valued, and have the following interior uniqueness property: Let , be single-valued elements, or branches, of and , defined on domains and , respectively, with . If and coincide on some set having at least one limit point , then and have the same domain of existence and coincide everywhere as complete analytic functions.
These formulations of uniqueness properties do not carry over to the case of functions of several complex variables , . For example, the analytic function is not identically zero, but vanishes on the analytic planes and of complex dimension . The following uniqueness properties hold for such functions:
1) If is an analytic function on a domain of the complex space and vanishes at all points of some non-empty open subset , then on .
2) If is an analytic function on a domain that vanishes at some point together with all its partial derivatives , ; ; , then on .
3) If is an analytic function on a domain that vanishes in a real neighbourhood of a point , that is, on a set , then on .
The difference between interior uniqueness properties in the cases and is caused by the different behaviour of power series in one and in several variables.
Boundary uniqueness properties.
The uniqueness theorem stated above for an analytic function of a single complex variable admits several generalizations to the case when the zeros of do not lie in the interior of the domain of analyticity, but on its boundary . The most general and deepest boundary uniqueness theorems were obtained by N.N. Luzin and I.I. Privalov in 1925 (cf. also Luzin–Privalov theorems). Let be a domain in the -plane bounded by a rectifiable curve , and let be a meromorphic function in . Let be a point of at which the tangent to exists; almost-all points of a rectifiable curve have this property. One says that has an angular boundary value at if tends to as tends to remaining within the intersection of the domain and the interior of any angle less than with vertex having the normal to at as its bisector.
The Luzin–Privalov boundary uniqueness theorem for angular boundary values then holds: If is meromorphic in a domain bounded by a rectifiable curve and takes angular boundary values zero on a set of positive Lebesgue measure, then . In general, a meromorphic function need not have boundary values on . But for a fairly large class of meromorphic functions, for example, for functions of bounded characteristic, the existence of angular boundary values almost everywhere on has been established.
Along with these, there are examples of bounded analytic functions in the unit disc that converge to zero in all senses on a given set of points of measure zero on the unit circle . Moreover, Luzin and Privalov have also constructed examples of analytic functions in the unit disc having zero radial boundary values, that is, tending to zero along radii, everywhere on a set of full measure . It turns out that in uniqueness problems the concept of the Baire category of a set is also very important. In fact, there is a Luzin–Privalov boundary uniqueness theorem for radial boundary values: If the function is meromorphic in the unit disc , has radial boundary values zero on a set situated on an arc of the unit circle , and if is metrically dense and of the second Baire category in , then . (A set is called metrically dense in if the intersection of with each subarc of has positive measure.)
The investigation of boundary uniqueness properties of analytic functions of several complex variables has not yet achieved the same degree of completeness as for functions of a single variable (see , ).
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Analogues of the Luzin–Privalov examples of holomorphic functions, not identically equal to zero and with radial boundary values zero almost everywhere, have been found for the unit ball in too, see [a1].
|[a1]||M. Hakim, N. Sibony, "Boundary properties of holomorphic functions in the ball in " Math. Ann. , 276 (1987) pp. 549–555|
Uniqueness properties of analytic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniqueness_properties_of_analytic_functions&oldid=12745