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.)
See also Boundary properties of analytic functions; Limit set.
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 [5], [6]).
References
[1] | A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) pp. Chapt. 3 (Translated from Russian) |
[2] | B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) pp. Chapt. 2 (In Russian) |
[3] | I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) |
[4] | E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) |
[5] | W. Rudin, "Function theory in the polydisc" , Benjamin (1969) |
[6] | G.M. [G.M. Khenkin] Henkin, E.M. [E.M. Chirka] Čirka, "Boundary properties of holomorphic functions of several complex variables" J. Soviet Math. , 5 (1976) pp. 612–687 Itogi Nauk. Sovrem. Probl. Mat. , 4 (1975) pp. 13–142 |
[7] | W. Rudin, "Function theory in the unit ball in ![]() |
[8] | P. Koosis, "Introduction to ![]() |
Comments
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].
References
[a1] | M. Hakim, N. Sibony, "Boundary properties of holomorphic functions in the ball in ![]() |
Uniqueness properties of analytic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniqueness_properties_of_analytic_functions&oldid=12745