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  " , Springer (1980) |
| [8] | P. Koosis, "Introduction to  -spaces. With an appendix on Wolff's proof of the corona theorem" , Cambridge Univ. Press (1980) |
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  " 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=49082