Talk:Function of several complex variables: Difference between revisions

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{{merged-from|Reinhardt ___domain|2020-11-20}}

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== I'm would like toconsidering merge athe Holomorphically___domain convexof hullholomorphy into Several complex variables, but the ___domain of holomorphy (theory) may be the title of this page ==

{{archive top|result=Withdraw[[User:Footlessmouse|Footlessmouse]] ([[User talk:Footlessmouse|talk]]) 11:05, 3 December 2020 (UTC)}}

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.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 07:55, 17 November 2020 (UTC)

The reason for branching into several complex variables from the trunk of complex analysis is that, unlike the case of one variable, the boundaries of all domains do not always become natural boundaries. I think the purpose of this page is to explain the mathematical elements that have become the elements that branch off from the trunk of complex analysis into several complex variables. This mathematical element seems to be called the ___domain of holomorphy, and since holomorphically convex and local Levi property etc. are conditions that make it a ___domain of holomorphy (And the theory of sheaf seems to be used to elucidate this condition. ), it seemed like we could read the ___domain of holomorphy as the theory that led to the branching of several complex variables from the trunk complex analysis. However, on the contrary, it seems good to rename the title of this page to the ___domain of holomorphy (theory). The lack of a page called several complex variables in [[Encyclopedia of Mathematics|EOM]] makes me think about this. The title of the textbook uses several complex variables, so my idea may be off the mark. thanks!--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 05:33, 21 December 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! [[User:Footlessmouse|Footlessmouse]] ([[User talk:Footlessmouse|talk]]) 08:59, 17 November 2020 (UTC)

I madeforgot ato draftsay ofit. theRedirecting leadto section.the Ititle thinkof thatthis thepage leadinstead sentenceof issection subjective,redirecting soto Ithe thought___domain Iof wouldholomorphy consultis beforealso addinga itsuggestion choice.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 1305:1539, 221 December 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!--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 09:55, 17 November 2020 (UTC)

:I don't think I agree with the assertion that several complex variables is distinct from complex analysis in one variable only in the sense that it is the study of domains of holomorphy/Stein manifolds. The interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and projective complex varieties for example, and has a different flavour to complex analytic geometry in <math>\mathbb{C}^n</math> or on Stein manifolds, which is what the current lead gives most of its weight to.

:::[[User:SilverMatsu|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! [[User:Footlessmouse|Footlessmouse]] ([[User talk:Footlessmouse|talk]]) 10:34, 17 November 2020 (UTC)

::::[[User:Footlessmouse|Footlessmouse]] Don’t worry about it. Thank you for contributing to the article with a wealth of knowledge.--[[User:SilverMatsu|SilverMatsu]] ([[User talk: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!--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 13:14, 18 November 2020 (UTC)

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

:I don't think it is standard anywhere to refer to the theory of complex functions of several variables as "domains of holomorphy theory" so Wikipedia should definitely avoid presenting that as the main name of the subject. As you point out however, people do refer to "several complex variables" and that is the title of the main book on the subject, so that is surely the better name for the article. I think we probably agree that the area is different enough in flavour to complex analysis of a single variable that it deserves its own article. I wouldn't be opposed to merging with [[complex analysis]], which is quite a thin article, but I don't think there is any particular need to. In particular I think complex analysis of a single variable gets a very different (and much broader) treatment pedagogically, and that is an article looked at frequently by people who are not pure mathematicians interested in several complex variables (engineers, physicists) and presenting all the definitions on that page in their largest generality would serve more to obfuscate the point rather than elucidate it for most visitors. Just my two cents. Thank you for improving the articles in complex analysis! [[User:Tazerenix|Tazerenix]] ([[User talk:Tazerenix|talk]]) 05:53, 23 December 2020 (UTC)

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

::Thank you very much for teaching me. Thanks for giving me many interesting examples of other manifolds. Sure, it seems too narrow, so I turn the suggestion into a section redirect to the ___domain of holomorphism. Then, I will modify the lead sentence and correct it to say, "One of the reasons why this field has come to be studied is that the boundary does not become a natural boundary." thanks!--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 08:01, 23 December 2020 (UTC)

::Withdraw from merge the ___domain of holomorphy. I might suggest merging conditions that are equivalent to the ___domain of holomorphy, but I thought that should be considered on the ___domain of holomorphy. Writing the ___domain of holomorphy on this page has no effect on withdrawal. thanks!--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 13:53, 26 December 2020 (UTC)

== A memo about the structure of the section ==

:{{ping|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. [[User:Footlessmouse|Footlessmouse]] ([[User talk:Footlessmouse|talk]]) 05:07, 2 December 2020 (UTC)

Please seeThe [[Infinitehttps://math.berkeley.edu/courses/choosing/course-dimensionaldescriptions#math212 holomorphy|thiscourse page]catalog]. Thanks!--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 1203:1419, 1121 DecemberJanuary 20202022 (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. [[User:Footlessmouse|Footlessmouse]] ([[User talk:Footlessmouse|talk]]) 05:25, 2 December 2020 (UTC)

== Is this right? ==

::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 <math>\Complex^n</math> 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 <math>\Complex^n \ (n\geq 2)</math>. 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! --[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 07:59, 3 December 2020 (UTC)

{{archive bottom}}

The section '''Stein manifold''' begins as follows:

=== Continued discussion ===

::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! --[[User:SilverMatsu|SilverMatsu]] ([[User talk: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. [[User:Footlessmouse|Footlessmouse]] ([[User talk: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. [[User:Footlessmouse|Footlessmouse]] ([[User talk:Footlessmouse|talk]]) 10:55, 3 December 2020 (UTC)

::::{{Ping|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! --[[User:SilverMatsu|SilverMatsu]] ([[User talk: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. --[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 13:12, 3 December 2020 (UTC)

"''Since a non-compact (open) Riemann surface always has a non-constant single-valued holomorphic function, and satisfies the [[second axiom of countability]], the open Riemann surface can be thought of ''1''-dimensional complex manifold to have a holomorphic embedding into a complex plane <math>\Complex</math>.''"

== I would like to merge the Reinhardt ___domain into Several complex variables ==

But wait. A holomorphic embedding is certainly also a topological embedding. But most non-compact surfaces do not have any topological embeddings into the complex plane.

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

:It seemed like it would work better to withdraw the integration and cite [http://encyclopediaofmath.org/index.php?title=Reinhardt_domain&oldid=48495|this page].The explanation that the ___domain is invariant due to rotation and the explanation of logarithmically-convex are more detailed.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 15:00, 19 November 2020 (UTC)

For instance, the torus with one point removed (the "punctured torus") has no topological embedding into the plane.

:{{ping|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) [[User:Footlessmouse|Footlessmouse]] ([[User talk:Footlessmouse|talk]]) 07:49, 20 November 2020 (UTC)

::{{ping|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.:)--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 08:06, 20 November 2020 (UTC)

Question: Is it possible that the phrase "holomorphic embedding" should be replaced with '''holomorphic immersion'''? Or just '''holomorphic mapping'''? [[Special:Contributions/2601:200:C000:1A0:9AE:98DB:C7E0:3910|2601:200:C000:1A0:9AE:98DB:C7E0:3910]] ([[User talk:2601:200:C000:1A0:9AE:98DB:C7E0:3910|talk]]) 17:57, 28 December 2022 (UTC)

== I'm would like to have a polydisc section ==

:Apparently Gunning and Narasimhan proved that every non-compact Riemann surface '''does''' in fact '''immerse''' in the complex plane.

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, <math>\Delta</math> for the open disk, <math>\overline{\Delta}</math> for the closed disk, and the ___domain mapped to the convex ___domain by the logarithmic transformation <math>\ln| z-a |</math> is <math>D^{*}</math> or <math>\Delta^{*}</math>. After discussing it, I will decide on the symbol that you use most often, so please give me your opinion. thanks.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 14:02, 25 November 2020 (UTC)

:(R. C. Gunning and Raghavan Narasimhan, Immersion of open Riemann surfaces, Math. Annalen 174 (1967), 103–108.)

:Therefore I will correct the text to reflect this fact. [[Special:Contributions/2601:200:C000:1A0:9AE:98DB:C7E0:3910|2601:200:C000:1A0:9AE:98DB:C7E0:3910]] ([[User talk:2601:200:C000:1A0:9AE:98DB:C7E0:3910|talk]]) 18:58, 28 December 2022 (UTC)

== Please say what you mean by superscript n ==

== Considering merge of Domain of holomorphy ==

The definition of a '''coherent sheaf''' is given as follows:

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.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 03:54, 28 November 2020 (UTC)

"''A '''coherent sheaf''' on a [[ringed space]] <math>(X, \mathcal O_X)</math> is a sheaf <math>\mathcal F</math> satisfying the following two properties:

== Proposed lead section ==

<ol type="1">

<li> <math>\mathcal F</math> is of ''finite type'' over <math>\mathcal O_X</math>, that is, every point in <math>X</math> has an [[open neighborhood]] <math>U</math> in <math>X</math> such that there is a surjective morphism <math>\mathcal{O}_X^n|_{U} \to \mathcal{F}|_{U} </math> for some natural number <math>n</math>;</li>

<li> for arbitrary open set <math>U\subseteq X</math>, arbitrary natural number <math>n</math>, and arbitrary morphism <math>\varphi: \mathcal{O}_X^n|_{U} \to \mathcal{F}|_{U} </math> of <math>\mathcal O_X</math>-modules, the kernel of <math>\varphi</math> is of finite type.''"</li></ol>

But what does the superscript n mean in the symbol "<math>\mathcal{O}_X^n|_{U}</math>"?

I made a draft of the lead section. I think that the lead sentence is subjective, so I thought I would consult before adding it.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 13:15, 2 December 2020 (UTC)

:I made changes before I read this, I am sorry about that. I like my first sentence better, but go ahead and make whatever changes you deem necessary, if anyone has a problem with it, we can discuss it then. Thanks! [[User:Footlessmouse|Footlessmouse]] ([[User talk:Footlessmouse|talk]]) 15:54, 2 December 2020 (UTC)

::Thank you for your reply! I like your modified sentence. I think I have to revise the sentence I wrote, but this page is more convenient for various editors to modify I thought, so I will add it. Thanks!--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 03:22, 3 December 2020 (UTC)

The article does not say, and nothing in the linked article [[ringed space]] uses this notation.

== Do you think several complex variables are functional theory or analysis? ==

I can guess two distinct possibilities for what "<math>\mathcal{O}_X^n|_{U}</math>" means.

Several complex variables start with Cauchy's integral formula, i.e. , the operation of integrating a function. The ___domain of holomorphy is the ___domain that is considered when an analytical operation is applied to a function, but in order to investigate the characteristics of the ___domain of holomorphy, methods in fields other than analysis are also used. However, since it is due to the integrate of functions, it is in the textbook of analysis. It doesn't seem to have anything to do with writing the article, but I'm interested so I'll ask you a question. I haven't been able to give an answer myself. --[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 14:58, 3 December 2020 (UTC)

I hope someone knowledgeable about this subject can explain the notation and avoid having thousands of future readers of Wikipedia also have to guess what it means. [[Special:Contributions/2601:200:C082:2EA0:E1A8:CCAE:61A1:827C|2601:200:C082:2EA0:E1A8:CCAE:61A1:827C]] ([[User talk:2601:200:C082:2EA0:E1A8:CCAE:61A1:827C|talk]]) 02:25, 9 February 2023 (UTC)

Addendum:　I searched the Wikipedia page, but according to the [[function theory]] page, it said "Theory of functions of a complex variable, the historical name for [[complex analysis]], the branch of mathematical analysis that investigates functions of complex numbers". Then the template on this page seems appropriate to change from a function to a complex analysis. If we can investigate the characteristics of complex variable functions by integral calculation, I think it is in the field of complex analysis. --[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 22:43, 3 December 2020 (UTC)

:From my experience, having taken only a single class in the subject as an undergrad, "complex analysis" is precisely defined by the second half of the DAB statement as "the branch of mathematical analysis that investigates functions of complex numbers", i.e. functions with at least one complex argument. My opinion on this is, lacking textbook consensus saying otherwise, unwaiverable. Though others may disagree and I do not own the page. From a pure linguistic point of view, it really doesn't make any sense to reserve "complex analysis" for the "study of functions of a single variable that is complex", it's too narrow of a field for such a broad term. [[User:Footlessmouse|Footlessmouse]] ([[User talk:Footlessmouse|talk]]) 07:46, 12 December 2020 (UTC)

::{{Ping|Footlessmouse}} Thank you for teaching me. If complex analysis is a branch of function (analysis) theory to complex numbers, I think it is clearer to say complex analysis. I also agree with your idea, as I think it's too narrow to limit to one variable. I think the complex analysis template theorem is too close to one variable. Where do you think you should talk? Thanks!--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 10:10, 12 December 2020 (UTC)

== Bad English and grammar ==

== I found a page that may be related to this page ==

Someone please proofread this article. [[Special:Contributions/2A02:6B6F:E98D:400:9BF:E37:5EAC:CB2C|2A02:6B6F:E98D:400:9BF:E37:5EAC:CB2C]] ([[User talk:2A02:6B6F:E98D:400:9BF:E37:5EAC:CB2C|talk]]) 22:16, 6 February 2025 (UTC)

Please see [[Infinite-dimensional holomorphy|this page]]. Thanks!--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 12:14, 11 December 2020 (UTC)

:I'm not sure this page is the best target for that page. That's a lot closer to [[holomorphic function]], IMO. Though it looks like it could use some work, and some references, either way. Thanks! [[User:Footlessmouse|Footlessmouse]] ([[User talk:Footlessmouse|talk]]) 07:48, 12 December 2020 (UTC)

::Thank you for your reply.I also seem to be close to a holomorphic function. When I looked at the page I introduced, it said "It is no longer true however that if a function is defined and holomorphic in a ball, its power series around the center of the ball is convergent in the entire ball; for example, there exist holomorphic functions defined on the entire space which have a finite radius of convergence". For Several complex variables, the Taylor expansion of the holomorphic function <math>f(z_1,\dots,z_n)</math> in the Reinhardt ___domain D, including the center a, has been shown to converge uniformly on any compact set in D, so I thought it might need to be covered on this page. My knowledge is inadequate and may not matter. My knowledge is inadequate, so it may be an unrelated topic. Thanks!--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 10:28, 12 December 2020 (UTC)