In
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 ...
, Hartogs's theorem is a fundamental result of
Friedrich Hartogs
Friedrich Moritz "Fritz" Hartogs (20 May 1874 – 18 August 1943) was a German-Jewish mathematician, known for his work on set theory and foundational results on several complex variables.
Life
Hartogs was the son of the merchant Gustav H ...
in 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 ...
. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if
is a function which is
analytic in each variable ''z''
''i'', 1 ≤ ''i'' ≤ ''n'', while the other variables are held constant, then ''F'' is a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
.
A
corollary
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
is that the function ''F'' is then in fact an analytic function in the ''n''-variable sense (i.e. that locally it has a
Taylor expansion). Therefore, 'separate analyticity' and 'analyticity' are coincident notions, in the theory of several complex variables.
Starting with the extra hypothesis that the function is continuous (or bounded), the theorem is much easier to prove and in this form is known as
Osgood's lemma In mathematics, Osgood's lemma, introduced by , is a proposition in complex analysis. It states that a continuous function of several complex variables that is holomorphic
In mathematics, a holomorphic function is a complex-valued function ...
.
There is no analogue of this
theorem for
real variables. If we assume that a function
is
differentiable (or even
analytic) in each variable separately, it is not true that
will necessarily be continuous. A counterexample in two dimensions is given by
:
If in addition we define
, this function has well-defined
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
s in
and
at the origin, but it is not
continuous at origin. (Indeed, the
limits along the lines
and
are not equal, so there is no way to extend the definition of
to include the origin and have the function be continuous there.)
References
*
Steven G. Krantz. ''Function Theory of Several Complex Variables'', AMS Chelsea Publishing, Providence, Rhode Island, 1992.
*
External links
*
{{PlanetMath attribution, urlname=HartogssTheoremOnSeparateAnalyticity, title=Hartogs's theorem on separate analyticity
Several complex variables
Theorems in complex analysis