Talk:Function of several complex variables/Archive 1

Vote for new external link

Here's my website full of example problems from complex variables. Someone please put it in the external links section if you think it's helpful!

http://www.exampleproblems.com/wiki/index.php/Complex_Variables

Not relevant, the link is about 1 variable, not several, like this article. Oleg Alexandrov (talk) 02:47, 29 December 2005 (UTC)

Agree with Oleg — not relevant. - Gauge 00:23, 8 January 2006 (UTC)

When several complex people agree on the same thing, it must be true! :) Oleg Alexandrov (talk) 02:32, 8 January 2006 (UTC)

Deformations of complex structures

On the sixth paragraph (the one that starts "from this points onwards there was a foundational theory...") mentions deformation theory of complex structures. Why is this on the foundations of the theory? I would find it quite interesting if a reason for the study of deformations of complex structures was given - and why it is considered to be one of the pillars (foundations) of the subject.

I would also find it quite nice if some applications of several complex variables to PDEs were mentioned.

I think that if this, quite nice, entry on several complex variables would include these two things, it would become even more enlightening. — Preceding unsigned comment added by 155.198.157.113 (talk • contribs) 18:57, 21 July 2006

Typesetting reversion

I reverted since there was an inconsistent mixture of typography: LaTeX and {{bigmath}}/{{math}} for inline and displayed formulae. The IP cleaned up the mixture. I'll check in detail if there were any typos the IP introduced in his/her edit, which may have motivated this reversion by Incnis Mrsi. M∧Ŝc²ħεИ_τlk 14:01, 17 June 2013 (UTC)

Could you look for IP’s user_talk before igniting an edit war? I reverted and will revert unconditionally an illegible crap like “−I”. The “imaginary unit i” also looks ugly, although I would not revert it were this one the only degradation. BTW, I do not see anything bad with “LaTeX and {{bigmath}}/{{math}} mixture of typography”: the purpose of {{math}} is namely to match appearance of <math> more closely than wiki code does. Incnis Mrsi (talk) 14:48, 17 June 2013 (UTC)

But the fonts in {{math}} are still different to LaTeX and people think it's jarring. I anticipated you'd say all this. Which is why, if there are no edit conflicts, I'll insert {{math}} uniformly inline leaving LaTeX displayed. M∧Ŝc²ħεИ_τlk 14:53, 17 June 2013 (UTC)

Also, consult WP:MOSMATH #Blackboard bold please. There are plenty of article where formatting is really poor. Why people like you and 99.241.86.114 start quarrels over typesetting preferences in relatively clean articles? Incnis Mrsi (talk) 14:59, 17 June 2013 (UTC)

I don't want to start an edit war, nor quarrel over my own preferences, just trying to to clean up the silly mix is all. You say this is a "relatively clean" article when LaTeX, HTML, and WP templates are used all over the place (LaTeX is inline and {{math}} displayed). As for the pointer to WP:MOSMATH, the article has a mix of Cⁿ and $\mathbb {C} ^{n}$, so tried to make them all consistent as blackboard bold - if others insist they could all be changed to bold after. M∧Ŝc²ħεИ_τlk 15:06, 17 June 2013 (UTC)

I do not obstruct any change which unambiguously improves at least something and does not make any demonstrable harm to the rest of code. Incnis Mrsi (talk) 15:16, 17 June 2013 (UTC)

I tried again in this edit, and assumed bold was the preference instead of blackboard bold for the real and complex number sets . M∧Ŝc²ħεИ_τlk 10:36, 18 June 2013 (UTC)

Too technical

I hope it's OK that I added the "technical" template at the top of the article. The reason is that the article doesn't give an overview that's comprehensible to someone who isn't already quite familiar with the subject. The lead is fine, but the next paragraph assumes intimate familiarity with the subject. The article could really do with a first section that introduces the subject to someone who's familiar with the prerequisites (e.g. single-variable complex analysis, multivariable real calculus) but who hasn't studied multivariable calculus per se. (Unfortunately I cannot write such a section, as I am a member of this target audience.) Nathaniel Virgo (talk) 13:05, 3 January 2015 (UTC)

You need to be still more specific. The paragraph following the lead is in the history section. Such a section is not even intended to be understood mathematically (for those unfamiliar with the subject). The section after that is on ℂⁿ. Is it here you get stuck? YohanN7 (talk) 16:40, 3 January 2015 (UTC)

I've moved the paragraph "The Cⁿ space" to the last, because it brings an example of Stein manifold but it needs a lots of efforts to reach it. Cousin problem, Levi problem, and the development of several complex variables must be explained, I think.--Enyokoyama (talk) 10:00, 23 March 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Function of several complex variables/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Needs splitting into sections, comparisons with single-variable case, examples, why they are "supposed to be" analytic, etc. Tompw 13:29, 20 November 2006 (UTC)

Last edited at 12:08, 16 April 2007 (UTC). Substituted at 02:35, 5 May 2016 (UTC)

Fix the ambiguous use of "these" and "they"

If the intended meaning of the first paragrah is that the theory of several complex variables studies only holomorphic functions then first sentence in the paragraph should say so. As the current article stands, the first sentence speaks of complex valued functions on complex n-tuples. The next sentence in the article says that "these functions" are "not just any functions". It is unclear which functions "these functions" refers to.

It would be clearer if the article began: "The theory of several complex variables deals with with a special type of complex valued functions on n-tuples of complex numbers. The functions of this special type are the holomorphic functions." - if that is the intended meaning.

Tashiro~enwiki (talk) 07:38, 23 June 2016 (UTC)

Series name

Considering the relationship with Cauchy's integral formula, I think it is good to use the page of Laurent expansion for explanation, but since it is a holomorphic point, I would like to describe it as Taylor expansion, Is there a good way?--SilverMatsu (talk) 12:53, 7 November 2020 (UTC)

Considering merge of Domain of holomorphy

For several complex variables, there are domains where the boundaries of the area do not become natural boundaries. Therefore, we will consider an domain where there is a function that is holomorphic and whose boundary is a natural boundary, but if we try to explain this on this page, the part that can be removed from the original page does not seem to be. On the contrary, if we omit the explanation of the domain of holomorphy, the evaluation of this page will be a stub except for the evaluation received by the part of the history of mathematics on this page. I feel my lack of ability, what I've added so far is still not enough to remove stubs from this page. First of all, I will aim for C evaluation.--SilverMatsu (talk) 03:54, 28 November 2020 (UTC)

I would like to merge the Reinhardt domain into Several complex variables

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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For the same reason as in the previous section.Oddly, it wasn't linked from this page.--SilverMatsu (talk) 12:08, 18 November 2020 (UTC)

It seemed like it would work better to withdraw the integration and cite page.The explanation that the domain is invariant due to rotation and the explanation of logarithmically-convex are more detailed.--SilverMatsu (talk) 15:00, 19 November 2020 (UTC)

@SilverMatsu: Hi! I wanted to let you know that if you come across a page that you wish to redirect, you can check the history of the page to see how active it is and potentially just do it. You can see WP:MERGE and WP:BLANK for instructions. The gist is, that if you do not expect anyone to complain, you do not need to open a discussion, you can just do it. If someone actually does object, they can simply undo the redirect of the page (but they would not remove content from here). So Reinhardt domain has been a problematic stub since 2009 and has no active editors - it's only touched once every couple of years. Therefore, there is no expectation that anyone will object to this one and you can just go ahead and do it without waiting for any kind of consensus. Also, when removing content, please put some kind of note in the edit summary explaining why (even if it says see talk page). You run a very high chance of having the edit reverted by the patrollers at WP:Recent changes patrol when you remove content without explanation. (Though they should see the page's templates and investigate further) Footlessmouse (talk) 07:49, 20 November 2020 (UTC)

@Footlessmouse: Hi! Thank you for teaching me. Since I have already written that several variables have points that converge outside the convergence circle, I added a section for the Reinhardt domain whose region is invariant due to rotation. I am very grateful that you are watching over my editing and giving me some advice.:)--SilverMatsu (talk) 08:06, 20 November 2020 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

I would like to merge a Holomorphically convex hull into Several complex variables

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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Holomorphically convex hulls are necessary to understand the properties that occur when increasing complex variables from one to several.So I wanted to add this topic to this page, but when I tried to add it, I noticed that the contents of the two pages were covered. I think the reason I came to think of the Holomorphically convex hull property was about the convergence region when the conditions described on the Holomorphically convex hull page were applied to the analytic function (holomorphic function).(Affects analytic continuation.) The power series is written on this page, so I would like to add a clause called Holomorphically convex hull on this page to integrate it. I would to leave the Holomorphically convex hull page as a redirect as a provisional. ( i.e. I do not object to increasing the content of the page and recreating a Holomorphically convex hull page.). thanks.--SilverMatsu (talk) 07:55, 17 November 2020 (UTC)

I completely changed my response here, sorry about that. This is fine with me, though I still believe it should all be merged into complex analysis. I have added templates that signify some of the problems I see with the article. In the eventuality that it is merged, many of them can be ignored. However, we do have to fix all the reference problems to make sure there is no original research. Thanks! Footlessmouse (talk) 08:59, 17 November 2020 (UTC)

Thank you for your feedback.I can't answer to the lead sentence because I didn't create it.The section on holomorphic functions supports your suggestion in the same part as in the case of one variable, and you may not need to distinguish it from one variable or several variables simply by saying complex analysis.On several variable pages, I added a note of order of integration because someone had already created a holomorphic function section, but personally, I think it's good to write on the complex analysis page.Unlike the case of one variable, there is no isolated zero of the holomorphic function, so the condition for establishing the identity theorem requires annotation. I think we need to consider a link to the identity theorem. Coherent sheaf, Hartogs' phenomenon, etc. are separate pages, so it seems that the reason why this page exists independently is weakening. I strongly agree that this page needs to be fix. I also understand that consolidating pages will increase the number of participants and improve the quality of the pages. Thanks!--SilverMatsu (talk) 09:55, 17 November 2020 (UTC)

SilverMatsu, that makes sense to me. I apologize for my slurry of comments. When I reread the original, I realized it did not have the right tone. It sort of seemed like I was yelling about it and being a little too aggressive, when you were certainly not the right person to yell at (not that anyone is). So my apologies. I appreciate the work you are doing, keep it up! Footlessmouse (talk) 10:34, 17 November 2020 (UTC)

Footlessmouse Don’t worry about it. Thank you for contributing to the article with a wealth of knowledge.--SilverMatsu (talk) 11:18, 17 November 2020 (UTC)

Added the contents of a Holomorphically convex hull under the analytic continuation section. I'll wait a little longer for discussion before deciding whether to redirect the original page. thanks!--SilverMatsu (talk) 13:14, 18 November 2020 (UTC)

The merge is complete. Thank you for your advice.--SilverMatsu (talk) 07:56, 25 November 2020 (UTC)

@Footlessmouse: Are you saying this whole article ought to be merged into Complex analysis? If so I would certainly disagree. Michael Hardy (talk) 04:51, 2 December 2020 (UTC)

@Michael Hardy: Would you mind elaborating? Do you actually believe Several complex variables is a distinct topic from complex analysis Or is page size the problem? The lead is patently false, attempting to define the topic as if it were totally separate from complex analysis. Also, I proposed all that when the page was a fraction of its current size, and so at the time size wasn't an issue, so if that's the problem, we can brain storm on that later. I'm not in any rush to do anything. I brought up many issues because this page was misleading and it would be more helpful for it to not be misleading, so laypeople do not learn incorrect information. At the time of proposing, I thought that merging with was the easiest way to accomplish that. Note, I have not reviewed all the recent changes, but appreciate the effort. Footlessmouse (talk) 05:07, 2 December 2020 (UTC)

Note, I would also support it being a stand alone article if, after cleaning up and properly adding citations to everything that needs it and deleting the OR, there is still enough material for it to merit its own page, but several parts of the article need to be rewritten for verifiability. Footlessmouse (talk) 05:25, 2 December 2020 (UTC)

There are some things that make merging difficult, and the definition of "a holomorphic function f in domain D" is not well-defined for several complex variables. i.e. Whether a holomorphic function can be defined depends on the range of domain D. Also, in the case of one variable, I took a point at infinity on the complex plane and considered a Riemann sphere. This is because the complex plane $\mathbb {C} ^{n}$ is connected, and the compact and connected Riemann surface was only the Riemann sphere, but there are innumerable analytic varieties that are compact and connected with the connection of $\mathbb {C} ^{n}\ (n\geq 2)$. However, the judgments differ depending on how you interpret the complex analysis page, so I think both opinions make sense. Adding a basic complex analysis (one-variable) textbook to this page may solve the citation problem. I haven't decided which textbook to use, but ... Thanks! --SilverMatsu (talk) 07:59, 3 December 2020 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Continued discussion

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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Addendum: In relation to the topic of citation, I reorganized the references and footnotes to make them easier to cite, but since I divided them into annotations and footnotes, I would like to used the ref tag in two sections, but but I can't. Also, since analytic continuation is not fully explained in how to write a lead statement, I am thinking of not mentioning analytic continuation at the time of the lead statement. Thanks! --SilverMatsu (talk) 08:52, 3 December 2020 (UTC)

I tried to change as little as possible in my edits, I just wanted to say it was part of complex analysis. As for the rest of it, I would have to read, I took a class on complex analysis, but it's been a while... I have fixed the footnotes, and regular references you add will automatically go in Footnotes while you can use ref name=note to add it to the annotations. I made a note in the edit summary. Footlessmouse (talk) 10:39, 3 December 2020 (UTC)

My bad, I understand better now, use template:harvnb to make add in citations that link to one of the books, but just use the other citations inline where the fact they corroborate appears in the page. You can give each of them a name using "name=" in their opening ref tag. I moved the footnotes to further reading for now and you can add them inline wherever they are, only because there are too many footnote sections and it was a little confusing. Footlessmouse (talk) 10:55, 3 December 2020 (UTC)

@Footlessmouse: Wow! The annotations and references section is now much easier to read. The talk page is also tidy and very convenient. Thank you very much! --SilverMatsu (talk) 12:29, 3 December 2020 (UTC)

The part of the lead sentence that I said earlier is the part that I added to the lead sentence, and I apologize for making ambiguous expressions while there were multiple editors who created the lead sentence. I thought the additions other than me were great, so I'm sorry if they seemed to oppose making the lead statement clear that they were part of complex analysis. Therefore, I would also like to cooperate in explaining that Several complex variables are areas within complex analysis, and as soon as I come up with a good method, I will continue to discuss this on the complex analysis talk page. --SilverMatsu (talk) 13:12, 3 December 2020 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

I'm would like to have a polydisc section

Writing the same domain over and over again makes the page hard to read, so I'd like to discuss it so that I can decide the symbol and write it concisely. Since wikipedia is edited by multiple people, the usage of symbols is different, and conversely, the symbols are not unified, so I think that it is difficult for anyone other than the first person to edit. For the time being, $\Delta$ for the open disk, ${\overline {\Delta }}$ for the closed disk, and the domain mapped to the convex domain by the logarithmic transformation $\ln |z-a|$ is $D^{*}$ or $\Delta ^{*}$. After discussing it, I will decide on the symbol that you use most often, so please give me your opinion. thanks.--SilverMatsu (talk) 14:02, 25 November 2020 (UTC)