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Keywords: 
Complex Analysis
Type of Math Object: 
Definition
Major Section: 
Reference
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Mathematics Subject Classification

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After doing a lot of reading, it seems clear that "domain" and "region" as used in complex analysis mean the same thing. Should I just delete the "region" entry and file a correction to the "domain" entry to that effect? Any other suggestions? There seems little sense in having two entries on the same thing.

Maybe this will start another interesting discussion like the one
about order of ring yesterday, but I prefer the use of the word
region for the reason that there is a potential for conflict with
"domnain". The problem, of course, is that domain means any set
on which a function is defined whether or not that set happens to
be open or closed or connected or disconnected.

Examples of where the definition of domain as connected and open
are troublesome arise quite naturally when discussing analytic
continuation. For instance, suppose we talk about anaytically
continuing the sine function from the positive real axis to the
complex plane. It is tempting to say that we have enlarged the
domain from the real axis to the complex plane, but this can be
confusing if we define "domain" to mean "open, connected subset
of the complex plane" because the real axis is a closed, as
opposed to an open subset of the complex plane. For another
example, we might start with a function, say the exponential
function, only defined on rational numbers and continue it to
the complex plane. In this case, the original domain of the
function is not even connected.

Trouble of a different sort arises when we make it to Riemann
surface theory. For instance, when we consider the rational
function f(z) = sqrt (1 + z^4), it is natural to define it as
a function on a torus. Now, while it may be closed, this torus
is not a subset of the complex plane, open or otherwise, but a
branched cover of it.

Maybe this is splitting hairs (but this is one of those bad hair
days where I have a whole bunch of split ends :) ) but the way that
I prefer to handle this is by convention rather than definition.
That is to say, I leave the definition of domain as a set on which
a function is defined alone but, when discussing complex analysis,
make the default assumption that all functions are holomorphic and
their domains are connected open subsets of the complex plane
(unless stated or implied otherwise).

When I want a conveeint term for "connected, open subset of the
complex plane", I use "region". The only possibity for conflict
I see there is with the colloquial sense of the term, but this is
not really an issue, especially since one can use "subset" in most
cases where one might use "region" colloquially.

As with the issue of order versus cardinality for rings, this
dilemma has similar historical origins. Around the same time
that group theory began, complex analysis also started (by many
of the same people). The terms "function" and "domain" then
acquired meanings peculiar to complex analysis. Fifty years
later, set theory came out and extended the meanings of the terms
"function" and "domain" to much more general situations. However,
by then the old meanings of the terms had already been well
established and got grandfathered in. While the use of function
to mean "holomorphic function" has faded away, the use of "domain"
to mean "region" is still with us.

Given that this usage is well-ingrained and likely to be encountered
by anyone reading a work of complex analysis, it definitely needs to
be mentioned here. I would suggest that you define "region", then
point out that, in complex analysis, the term "domain" is often used
as a synonym, but that this usage, though quite common, has the
potential to conflict with the more modern set-theoretic definition
of "domain", so caveat lector!

I have decided to edit region in response to the post to which I am replying.

I have found no evidence of "region" being used to refer to a nonempty open connected subset of C^n, whereas the PM entry for "domain" claims that "domain" *is* used for this purpose. If anyone has any information about this, please let me know.

I appreciate all of this historical background Ray. I especially enjoy the timeline of what type of mathematics came into being when and how that has affected the vocabulary (which we have already discussed for order vs cardinality and domain vs region). I really think that PM needs encyclopedia entries on the history of mathematics that includes information such as provided in your most recent posts.

Warren

I think there exists a non-ambiguous term meaning the 'set in which a fuction is defined', namely the "definition set". But it is quite rare.
Jussi

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