In mathematics, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of

several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...

. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if

F:^n \to

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. 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 In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...

). 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. There is no analogue of this

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...

for real variables. If we assume that a function

f \colon ^n \to

differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point i ...

(or even analytic) in each variable separately, it is not true that

f

will necessarily be continuous. A counterexample in two dimensions is given by :

f(x,y) = \frac.

If in addition we define

f(0,0)=0

, 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). Pa ...

s in

x

and

y

at the origin, but it is not

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

at origin. (Indeed, the

limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

along the lines

x=y

and

x=-y

are not equal, so there is no way to extend the definition of

f

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