Talk:Domain (mathematical analysis)

Dedication

This page is dedicated to the memory of Oded Schramm, wikipedian and scientist.

Exterior Domain

"Exterior, or external, domain is the complement of a bounded domain." Thus, an exterior domain is closed? — Preceding unsigned comment added by 67.220.7.72 (talk) 01:23, 5 November 2012 (UTC)

Simply-Connected or Just Connected?

It seems like most definitions in books use simply-connectedness for a domain, not just connectedness.

Merge discussion (Domain and Region)

The differences between these terms are subtle and I think we need to explain both on one page. See Wikipedia talk:WikiProject Mathematics/Archive/2021/Apr#Is it better to avoid just writing domain in articles? for other articles that need merging.--SilverMatsu (talk) 01:53, 15 September 2021 (UTC)

Seems like an okay idea to me (disclaimer: I am not a mathematician). Seems like in complex analysis at least, the two words are both used for connected open sets, with some sources picking one and some sources picking the other. –jacobolus (t) 17:48, 19 September 2021 (UTC)

Thank you for your reply. If either name doesn't take precedence, I think it might be better to rename this article to "Domain and Region".--SilverMatsu (talk) 08:17, 20 September 2021 (UTC)

@Jacobolus: Region (mathematics) changed the page name to Region (mathematical analysis) because it explained mathematical analysis.--SilverMatsu (talk) 16:12, 20 September 2021 (UTC)

I think you should be WP:BOLD and merge the articles into Domain, list both names as synonyms, and then explain in the article that choice of term and precise definitions vary from author to author. If you feel ambitious you can try to gather up a list of influential textbooks which picked one term or the other. –jacobolus (t) 17:12, 20 September 2021 (UTC)

If you want you could make a note at Wikipedia talk:WikiProject Mathematics. –jacobolus (t) 17:15, 20 September 2021 (UTC)

Thank you your reply. Added encyclopedia of mathematics and planet math to references. For complex coordinate space, I'm thinking of using blackboard bold. Because it seems confusing with a smooth function.--SilverMatsu (talk) 16:00, 22 September 2021 (UTC)

Okay, I went ahead and redirected Region -> Domain, after adding the three references from the Region article. I don’t expect anyone is going to mind having these merged. –jacobolus (t) 05:15, 23 September 2021 (UTC)

Thank you! I've added a template at the top of this page.--SilverMatsu (talk) 06:20, 23 September 2021 (UTC)

Burkhardt

Hi SilverMatsu. I don’t read German. Maybe you’ll have an easier time figuring out the various definitions used in Burkhardt (1897) – https://archive.org/details/einfhrungindiet01burkgoog/page/n92/ – There is an English translation (1913) of the 4th edition here – https://archive.org/details/theoryfunctions00rasorich/page/n157/mode/2up – which defines a region as a connected closed set containing inner points and a domain as a surface bounded by a path. –jacobolus (t) 19:52, 25 September 2021 (UTC)

Clarification

Hi. I find the following sentence unclear: "This is a different concept than the domain of a function, though it is often used for that purpose..." Can we try and clarify the relationship between the two concepts? Would it be right to say the mathematical-analysis concept is more general, and the domain of a function is an application or special case restricted to the input of functions? If so, then "different" is not strictly valid and would seem to be wrong in the present version. How about saying, instead: "When a given domain or region is used to define the inputs accepted by a function, it is called the domain of a function". fgnievinski (talk) 17:04, 9 March 2023 (UTC)

No, the domain of a function is the set of possible "inputs" to the function. Any arbitrary set can be the domain of some function.

This sense of domain is a non-empty connected open set inside a topological space (often $\mathbb {C}$ or $\mathbb {C} ^{n}$).

Often a domain (in this sense) is used as the domain of a function. But they are distinct concepts. I don’t think there’s really any easy way to “clarify” this without extended elaboration, which would certainly not be in scope for the lead section of this article. I think readers will just have to click through to the 'domain of a function' wiki-link and compare the definitions for themselves. –jacobolus (t) 18:06, 9 March 2023 (UTC)

In section Domain of a function#Other uses, it only says: "The word 'domain' is used with other related meanings in some areas of mathematics." That section has ample space for clarifying how the two concepts are related. Currently there's only a restatement of the various definitions of Domain (mathematical analysis). But there's nothing about their relationship. fgnievinski (talk) 18:58, 9 March 2023 (UTC)

The definitions seem clear enough for anyone who knows the jargon used (or willing to click through the wiki-links to figure out what the jargon means). The form there is: "In X context term means «definition X». In Y context term means «definition Y». In Z context term means «definition Z». There’s not really an inherent “relationship” between the different definitions. But if you want you can add a sentence that «an X term can sometimes be a Y term».

Or feel free to write another paragraph there elaborating. You can add some diagrams or whatever if you think it helps. –jacobolus (t) 19:52, 9 March 2023 (UTC)

Alright, I tried to clean up that section a bit. Does that help? –jacobolus (t) 20:12, 9 March 2023 (UTC)

@Jacobolus: Thank you very much for the improvement in section Domain of a function#Other uses. The way it was before, it could give the wrong impression the terms were homonyms for completely difference concepts. Now their not-so-mysterious relationship was clarified. Thanks also to Rgdboer for improving the lead of Domain (mathematical analysis) in a simple and effective way. fgnievinski (talk) 07:18, 11 March 2023 (UTC)