In
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include converg ...
and
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, branches of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the identity theorem for
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
s states: given functions ''f'' and ''g'' analytic on a
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
''D'' (open and connected subset of
or
), if ''f'' = ''g'' on some
, where
has an
accumulation point, then ''f'' = ''g'' on ''D''.
Thus an analytic function is completely determined by its values on a single open neighborhood in ''D'', or even a countable subset of ''D'' (provided this contains a converging sequence). This is not true in general for real-differentiable functions, even
infinitely real-differentiable functions. In comparison, analytic functions are a much more rigid notion. Informally, one sometimes summarizes the theorem by saying analytic functions are "hard" (as opposed to, say, continuous functions which are "soft").
The underpinning fact from which the theorem is established is the
expandability of a holomorphic function into its Taylor series.
The connectedness assumption on the domain ''D'' is necessary. For example, if ''D'' consists of two disjoint
open sets
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
,
can be
on one open set, and
on another, while
is
on one, and
on another.
Lemma
If two holomorphic functions
and
on a domain ''D'' agree on a set S which has an accumulation point
in
, then
on a disk in
centered at
.
To prove this, it is enough to show that
for all
.
If this is not the case, let
be the smallest nonnegative integer with
. By holomorphy, we have the following Taylor series representation in some open neighborhood U of
:
:
By continuity,
is non-zero in some small open disk
around
. But then
on the punctured set
. This contradicts the assumption that
is an accumulation point of
.
This lemma shows that for a complex number
, the
fiber
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
is a discrete (and therefore countable) set, unless
.
Proof
Define the set on which
and
have the same Taylor expansion:
We'll show
is nonempty,
open, and closed. Then by
connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be s ...
of
,
must be all of
, which implies
on
.
By the lemma,
in a disk centered at
in
, they have the same Taylor series at
, so
,
is nonempty.
As
and
are holomorphic on
,
, the Taylor series of
and
at
have non-zero
radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
. Therefore, the open disk
also lies in
for some
. So
is open.
By holomorphy of
and
, they have holomorphic derivatives, so all
are continuous. This means that
is closed for all
.
is an intersection of closed sets, so it's closed.
Full characterisation
Since the Identity Theorem is concerned with the equality of two
holomorphic functions
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
, we can simply consider the difference (which remains holomorphic) and can simply characterise when a holomorphic function is identically
. The following result can be found in.
Claim
Let
denote a non-empty,
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
open subset of the complex plane.
For
the following are equivalent.
#
on
;
# the set
contains an
accumulation point,
;
# the set
is non-empty, where
.
Proof
The directions (1
2) and (1
3) hold trivially.
For (3
1), by connectedness of
it suffices to prove that the non-empty subset,
, is clopen (since a topological space is connected if and only if it has no proper clopen subsets).
Since holomorphic functions are infinitely differentiable, ''i.e.''
, it is clear that
is closed. To show openness, consider some
.
Consider an open ball
containing
, in which
has a convergent Taylor-series expansion centered on
.
By virtue of
, all coefficients of this series are
, whence
on
.
It follows that all
-th derivatives of
are
on
, whence
.
So each
lies in the interior of
.
Towards (2
3), fix an accumulation point
.
We now prove directly by induction that
for each
.
To this end let
be strictly smaller than the convergence radius of the power series expansion of
around
, given by
. Fix now some
and assume that
for all
. Then for
manipulation of the power series expansion yields
Note that, since
is smaller than radius of the power series, one can readily derive that the power series
is continuous and thus bounded on
.
Now, since
is an accumulation point in
, there is a sequence of points
convergent to
.
Since
on
and since each
, the expression in () yields
By the boundedness of
on
,
it follows that
, whence
.
Via induction the claim holds.
Q.E.D.
See also
*
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
*
Identity theorem for Riemann surfaces In mathematics, the identity theorem for Riemann surfaces is a theorem that states that a holomorphic function is completely determined by its values on any subset of its domain that has a limit point.
Statement of the theorem
Let X and Y be Rie ...
References
* {{cite book
, author1=Ablowitz, Mark J.
, author2=Fokas A. S.
, language=en
, title=Complex variables: Introduction and applications
, publisher=Cambridge University Press
, pages=122
, location=Cambridge, UK
, year=1997
, isbn=0-521-48058-2
Theorems_in_real_analysis
Theorems in complex analysis
Articles containing proofs