In the theory of functions of
several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
, Hartogs's extension theorem is a statement about the
singularities of
holomorphic function
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 deriv ...
s of several variables. Informally, it states that the
support
Support may refer to:
Arts, entertainment, and media
* Supporting character
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Support
Construction
* Support (structure), or lateral support, a ...
of the singularities of such functions cannot be
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
, therefore the singular set of a function of several complex variables must (loosely speaking) 'go off to infinity' in some direction. More precisely, it shows that an
isolated singularity is always a
removable singularity
In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbour ...
for any
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 ...
of complex variables. A first version of this theorem was proved by
Friedrich Hartogs,
[See the original paper of and its description in various historical surveys by , and . In particular, in this last reference on p. 132, the Author explicitly writes :-"''As it is pointed out in the title of , and as the reader shall soon see, the key tool in the proof is the Cauchy integral formula''".] and as such it is known also as Hartogs's lemma and Hartogs's principle: in earlier
Soviet
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, ...
literature, it is also called Osgood–Brown theorem, acknowledging later work by
Arthur Barton Brown and
William Fogg Osgood
William Fogg Osgood (March 10, 1864, Boston – July 22, 1943, Belmont, Massachusetts) was an American mathematician.
Education and career
In 1886, he graduated from Harvard, where, after studying at the universities of Göttingen (1887–1889 ...
. This property of holomorphic functions of several variables is also called
Hartogs's phenomenon: however, the locution "Hartogs's phenomenon" is also used to identify the property of solutions of
systems
A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and expresse ...
of
partial differential or
convolution equations satisfying Hartogs type theorems.
Historical note
The original proof was given by
Friedrich Hartogs in 1906, using
Cauchy's integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary ...
for
functions of several complex variables.
Today, usual proofs rely on either the
Bochner–Martinelli–Koppelman formula or the solution of the inhomogeneous
Cauchy–Riemann equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differenti ...
with compact support. The latter approach is due to
Leon Ehrenpreis
Eliezer 'Leon' Ehrenpreis (May 22, 1930 – August 16, 2010, Brooklyn) was a mathematician at Temple University who proved the Malgrange–Ehrenpreis theorem, the fundamental theorem about differential operators with constant coefficients. He p ...
who initiated it in the paper . Yet another very simple proof of this result was given by
Gaetano Fichera
Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables. He was born in Acireale, and died in Rome.
Biog ...
in the paper , by using his solution of the
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.
The Dirichlet prob ...
for
holomorphic function
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 deriv ...
s of several variables and the related concept of
CR-function: later he extended the theorem to a certain class of
partial differential operators in the paper , and his ideas were later further explored by Giuliano Bratti. Also the Japanese school of the theory of
partial differential operators worked much on this topic, with notable contributions by Akira Kaneko. Their approach is to use
Ehrenpreis's fundamental principle In mathematical analysis, Ehrenpreis's fundamental principle, introduced by Leon Ehrenpreis, states:
:Every solution of a system (in general, overdetermined) of homogeneous partial differential equation
In mathematics, a partial differential equ ...
.
Hartogs's phenomenon
For example, in two variables, consider the interior domain
:
in the two-dimensional polydisk
where
.
Theorem : any holomorphic functions
on
are analytically continued to
. Namely, there is a holomorphic function
on
such that
on
.
Such a phenomenon is called Hartogs's phenomenon, which lead to the notion of this Hartogs's extension theorem and the
domain of holomorphy
In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be extended to a bigger domain.
For ...
.
Formal statement and proof
:Let be a
holomorphic function
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 deriv ...
on a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
, where is an open subset of () and is a compact subset of . If the
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
is connected, then can be extended to a unique holomorphic function on .
Ehrenpreis' proof is based on the existence of smooth
bump function
In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bum ...
s, unique continuation of holomorphic functions, and the
Poincaré lemma — the last in the form that for any smooth and compactly supported differential (0,1)-form on with , there exists a smooth and compactly supported function on with . The crucial assumption is required for the validity of this Poincaré lemma; if then it is generally impossible for to be compactly supported.
The ansatz for is for smooth functions and on ; such an expression is meaningful provided that is identically equal to zero where is undefined (namely on ). Furthermore, given any holomorphic function on which is equal to on ''some'' open set, unique continuation (based on connectedness of ) shows that it is equal to on ''all'' of .
The holomorphicity of this function is identical to the condition . For any smooth function , the differential (0,1)-form is -closed. Choosing to be a smooth function which is identically equal to zero on and identically equal to one on the complement of some compact subset of , this (0,1)-form additionally has compact support, so that the Poincaré lemma identifies an appropriate of compact support. This defines as a holomorphic function on ; it only remains to show (following the above comments) that it coincides with on some open set.
On the set , is holomorphic since is identically constant. Since it is zero near infinity, unique continuation applies to show that it is identically zero on some open subset of .
[Any connected component of must intersect in a nonempty open set. To see the nonemptiness, connect an arbitrary point of to some point of via a line. The intersection of the line with may have many connected components, but the component containing gives a continuous path from into .] Thus, on this open subset, equals and the existence part of Hartog's theorem is proved. Uniqueness is automatic from unique continuation, based on connectedness of .
Counterexamples in dimension one
The theorem does not hold when . To see this, it suffices to consider the function , which is clearly holomorphic in but cannot be continued as a holomorphic function on the whole . Therefore, the Hartogs's phenomenon is an elementary phenomenon that highlights the difference between the theory of functions of one and several complex variables.
Notes
References
Historical references
*.
*.
*. A historical paper correcting some inexact historical statements in the theory of
holomorphic functions of several variables, particularly concerning contributions of
Gaetano Fichera
Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables. He was born in Acireale, and died in Rome.
Biog ...
and
Francesco Severi
Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician. He was the chair of the committee on Fields Medal on 1936, at the first delivery.
Severi was born in Arezzo, Italy. He is famous for his contributions to algebr ...
.
*. This is the first paper where a general solution to the
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.
The Dirichlet prob ...
for
pluriharmonic functions is given for general
real analytic data on a real analytic
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
. A translation of the title reads as:-"''Solution of the general Dirichlet problem for biharmonic functions''".
*. A translation of the title is:-"''Lectures on analytic functions of several complex variables – Lectured in 1956–57 at the Istituto Nazionale di Alta Matematica in Rome''". This book consist of lecture notes from a course held by Francesco Severi at the
Istituto Nazionale di Alta Matematica (which at present bears his name), and includes appendices of
Enzo Martinelli
Enzo Martinelli (11 November 1911 – 27 August 1999 writes that his death year is 1998, unlike to , and , but it is probably a typographical error.) was an Italian mathematician, working in the theory of functions of several complex variable ...
,
Giovanni Battista Rizza
Giovanni Battista Rizza (7 February 1924 – 15 October 2018), officially known as Giambattista Rizza, was an Italian mathematician, working in the fields of complex analysis of several variables and in differential geometry: he is known for hi ...
and
Mario Benedicty.
*.
* (
Zentralblatt
zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastruct ...
review of the original
Russian edition). One of the first modern monographs on the theory of
several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
, being different from other ones of the same period due to the extensive use of
generalized function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
s.
Scientific references
*.
*.
*
*
*
*
*. A fundamental paper in the theory of Hartogs's phenomenon. The typographical error in the title is reproduced as it appears in the original version of the paper.
*. An epoch-making paper in the theory of
CR-functions, where the Dirichlet problem for
analytic functions of several complex variables is solved for general data. A translation of the title reads as:-"''Characterization of the trace, on the boundary of a domain, of an analytic function of several complex variables''".
*. An English translation of the title reads as:-"''Hartogs phenomenon for certain linear partial differential operators''".
*. Available at th
SEALS Portal
* (see also , the cumulative review of several papers by E. Trost). Available at th
SEALS Portal
*.
*. Available at th
DigiZeitschriften
*.
*, available a
Project Euclid
*. Available at th
SEALS Portal
*.
*. An English translation of the title reads as:-"''A fundamental property of the domain of holomorphy of an analytic function of one real variable and one complex variable''".
*. Available at th
SEALS Portal
External links
*
*
*
*{{PlanetMath, urlname=ProofOfHartogsTheorem, title=Proof of Hartogs' theorem
Several complex variables
Theorems in complex analysis