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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  $  D $
be a domain in the complex plane  $  \mathbf C = \mathbf C  ^ {1} $. 
The classical interior uniqueness theorem for holomorphic (that is, single-valued analytic) functions on  $  D $
states that if two holomorphic functions  $  f ( z) $
and  $  g ( z) $
in  $  D $
coincide on some set  $  E \subset  D $
containing at least one limit point in  $  D $, 
then  $  f ( z) \equiv g ( z) $
everywhere in  $  D $. 
In other words, if a holomorphic function  $  f ( z) $
in  $  D $
vanishes on a set  $  E \subset  D $
having at least one limit point in  $  D $, 
then  $  f ( z) \equiv 0 $. 
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  $  z $. 
The uniqueness property remains valid for meromorphic functions  $  f ( z) $
and  $  g ( z) $
in  $  D $
if one regards the poles of  $  f ( z) $
and  $  g ( z) $
as points at which the functions take the value  $  \infty $.

In particular, if two analytic functions  $  f ( z) $
and  $  g ( z) $
coincide in an arbitrarily small neighbourhood of some point or on an arbitrarily small arc of some continuous curve, then  $  f ( z) \equiv g ( z) $. 
Another corollary: The set of  $  A $-
points of an analytic function  $  f ( z) $, 
that is, the set of points  $  z $
at which  $  f ( z) = A $, 
cannot have limit points inside the domain of definition  $  D $, 
provided that  $  f ( z) \not\equiv A $.

Two complete analytic functions in the sense of Weierstrass (cf. [[Complete analytic function|Complete analytic function]])  $  F ( z) $, 
$  G ( z) $
are, in general, multiple valued, and have the following interior uniqueness property: Let  $  f ( z) $, 
$  g ( z) $
be single-valued elements, or branches, of  $  F ( z) $
and  $  G ( z) $, 
defined on domains  $  D _ {1} $
and  $  D _ {2} $, 
respectively, with  $  D _ {1} \cap D _ {2} \neq \emptyset $. 
If  $  f ( z) $
and  $  g ( z) $
coincide on some set  $  E \subset  D _ {1} \cap D _ {2} $
having at least one limit point  $  z _ {0} \in D _ {1} \cap D _ {2} $, 
then  $  F ( z) $
and  $  G ( z) $
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  $  f ( z) $
of several complex variables  $  z = ( z _ {1} \dots z _ {n} ) $, 
$  n > 1 $. 
For example, the analytic function  $  f ( z) = z _ {1} z _ {2} $
is not identically zero, but vanishes on the analytic planes  $  z _ {1} = 0 $
and  $  z _ {2} = 0 $
of complex dimension  $  n - 1 $. 
The following uniqueness properties hold for such functions:

1) If  $  f ( z) $
is an analytic function on a domain  $  D $
of the complex space  $  \mathbf C  ^ {n} $
and vanishes at all points of some non-empty open subset  $  U \subset  D $, 
then  $  f ( z) \equiv 0 $
on  $  D $.

2) If  $  f ( z) $
is an analytic function on a domain  $  D \subset  \mathbf C  ^ {n} $
that vanishes at some point  $  z  ^ {0} \in D $
together with all its partial derivatives  $  \partial   ^ {k} f / \partial  z _ {1} ^ {k _ {1} } \dots \partial  z _ {n} ^ {k _ {n} } $, 
$  k = k _ {1} + \dots + k _ {n} $; 
$  k _ {j} = 0 , 1 ,\dots $; 
$  j = 1 \dots n $, 
then  $  f ( z) \equiv 0 $
on  $  D $.

3) If  $  f ( z) $
is an analytic function on a domain  $  D \subset  \mathbf C  ^ {n} $
that vanishes in a real neighbourhood  $  U _  \partial  $
of a point  $  z  ^ {0} = x  ^ {0} + iy  ^ {0} \in D $, 
that is, on a set  $  U _  \partial  = \{ {z = x + iy \in \mathbf C  ^ {n} } : {| x - x  ^ {0} | < r,  y = y  ^ {0} } \} $, 
then  $  f ( z) \equiv 0 $
on  $  D $.

The difference between interior uniqueness properties in the cases  $  n = 1 $
and  $  n > 1 $
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  $  f ( z) $
of a single complex variable admits several generalizations to the case when the zeros of  $  f ( z) $
do not lie in the interior of the domain  $  D $
of analyticity, but on its boundary  $  \Gamma $. 
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|Luzin–Privalov theorems]]). Let  $  D $
be a domain in the  $  z $-
plane bounded by a rectifiable curve  $  \Gamma $, 
and let  $  f ( z) $
be a meromorphic function in  $  D $. 
Let  $  \zeta _ {0} $
be a point of  $  \Gamma $
at which the tangent to  $  \Gamma $
exists; almost-all points of a rectifiable curve have this property. One says that  $  f ( z) $
has an angular boundary value  $  A $
at  $  \zeta _ {0} $
if  $  f ( z) $
tends to  $  A $
as  $  z $
tends to  $  \zeta _ {0} $
remaining within the intersection of the domain  $  D $
and the interior of any angle less than  $  \pi $
with vertex  $  \zeta _ {0} $
having the normal to  $  \Gamma $
at  $  \zeta _ {0} $
as its bisector.

The Luzin–Privalov boundary uniqueness theorem for angular boundary values then holds: If  $  f ( z) $
is meromorphic in a domain  $  D $
bounded by a rectifiable curve  $  \Gamma $
and takes angular boundary values zero on a set  $  E \subset  \Gamma $
of positive Lebesgue measure, then  $  f ( z) \equiv 0 $. 
In general, a meromorphic function need not have boundary values on  $  \Gamma $. 
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  $  \Gamma $
has been established.

Along with these, there are examples of bounded analytic functions in the unit disc  $  D $
that converge to zero in all senses on a given set of points  $  E $
of measure zero on the unit circle  $  \Gamma $. 
Moreover, Luzin and Privalov have also constructed examples of analytic functions in the unit disc  $  D $
having zero radial boundary values, that is, tending to zero along radii, everywhere on a set  $  E \subset  \Gamma $
of full measure  $  2 \pi $. 
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  $  f ( z) $
is meromorphic in the unit disc  $  D $, 
has radial boundary values zero on a set  $  E $
situated on an arc  $  \sigma $
of the unit circle  $  \Gamma $, 
and if  $  E $
is metrically dense and of the second Baire category in  $  \sigma $, 
then  $  f ( z) \equiv 0 $. 
(A set  $  E $
is called metrically dense in  $  \sigma $
if the intersection of  $  E $
with each subarc of  $  \sigma $
has positive measure.)

See also [[Boundary properties of analytic functions|Boundary properties of analytic functions]]; [[Limit set|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 [[#References|[5]]], [[#References|[6]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  pp. Chapt. 3  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,   "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  pp. Chapt. 2  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,   "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.F. Collingwood,   A.J. Lohwater,   "The theory of cluster sets" , Cambridge Univ. Press  (1966)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Rudin,   "Function theory in the polydisc" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  W. Rudin,   "Function theory in the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095430/u095430128.png" />" , Springer  (1980)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  P. Koosis,   "Introduction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095430/u095430129.png" />-spaces. With an appendix on Wolff's proof of the corona theorem" , Cambridge Univ. Press  (1980)</TD></TR></table>

====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  $  \mathbf C  ^ {n} $
too, see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hakim,   N. Sibony,   "Boundary properties of holomorphic functions in the ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095430/u095430131.png" />"  ''Math. Ann.'' , '''276'''  (1987)  pp. 549–555</TD></TR></table>