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 &mdash; 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 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, which may have motivated by Incnis Mrsi. M&and;Ŝc2ħεИτlk 14:01, 17 June 2013 (UTC)
 * Could you look for IP’s user_talk before igniting an edit war? 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 &#123;{math}} is namely to match appearance of &lt;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&and;Ŝc2ħεИτlk 14:53, 17 June 2013 (UTC)
 * Also, consult WP:MOSMATH 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&and;Ŝc2ħεИτlk 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, and assumed bold was the preference instead of blackboard bold for the real and complex number sets. M&and;Ŝc2ħεИτlk 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 $ℂ^{n}$. 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
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
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 |this 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)


 * 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)
 * 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)

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

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)


 * 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 $$\Complex^n$$ 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 $$\Complex^n \ (n\geq 2)$$. 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)

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! --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)
 * 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)

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)

Proposed lead section
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.--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! 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!--SilverMatsu (talk) 03:22, 3 December 2020 (UTC)

Do you think several complex variables are functional theory or analysis?
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. --SilverMatsu (talk) 14:58, 3 December 2020 (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. --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. Footlessmouse (talk) 07:46, 12 December 2020 (UTC)
 * 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!--SilverMatsu (talk) 10:10, 12 December 2020 (UTC)
 * Sorry, I'm not sure what you mean with your last statement and question. Randomly, though, I found this that may actually help both of us understand better. article on JSTOR titled "What is several complex variables" by Steven G. Krantz. Because he is an established expert and it is published in a reliable source, you can use that as a reference when talking about the differences. Footlessmouse (talk) 11:16, 12 December 2020 (UTC)
 * Thank you for giving me a reliable reference. I Make time to read. I'm sorry. The name of the template was incorrect. The correct name was Template:Complex analysis sidebar.--SilverMatsu (talk) 11:41, 12 December 2020 (UTC)

It may be better to say that this page is a theory of Several complex variables function rather than a function theory of Several complex variables. If the analysis part of this page gets too big, it seems that it can be divided into function theory of Several complex variables. I'll look at the redirects on this page. thanks!--SilverMatsu (talk) 06:20, 24 December 2020 (UTC)
 * Addendum:Therefore, templates seem to be better for functions than complex analysis.--SilverMatsu (talk) 06:22, 24 December 2020 (UTC)

I found a page that may be related to this page
Please see this page. Thanks!--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! 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 $$f(z_1,\dots,z_n)$$ on the Reinhardt domain D, including the center a, has been shown to converge uniformly on any compact set on 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!--SilverMatsu (talk) 10:28, 12 December 2020 (UTC)
 * My knowledge is also inadequate, hopefully a mathematician can look over all this at some point in the near future. My best advice is that while you are rewriting large chunks of the page, you should just follow what established, reliable sources say. If they are all talking about a concept, then it should be mentioned or summarized here, otherwise you can probably get away without mentioning at all. In the meanwhile, you can add it to "See also". Footlessmouse (talk) 11:19, 12 December 2020 (UTC)
 * Thank you for your advice. I will add it to the See also.--SilverMatsu (talk) 11:44, 12 December 2020 (UTC)
 * Looking at the example of Compact space, it seems that there is an example of a bounded closed set i.e. unit ball that does not become compact in infinite dimensions. I think I missed the condition of compact set. I also likely need to read the references on the page where the example is shown.--SilverMatsu (talk) 12:20, 12 December 2020 (UTC)

I changed the reference link of the infinite-dimension page, so it should be available for download. Thanks to Mike Turnbull advice.--SilverMatsu (talk) 13:41, 14 December 2020 (UTC)

I was able to find out the weak holomorphic.

Weak definition
 * A function $$h:D\rightarrow Y$$ is holomorphic if it is locally bounded and if for each $$x\in D$$, $$y\in X$$ and linear functional $$\ell\in Y^{\ast}$$, the function $$f(\lambda)=\ell (h(x+\lambda y))$$ is holomorphic at $$\lambda=0$$.

Since it says useful criterion, the holomorphic on this page may mean a weak holomorphic. I've read that the reason why holomorphy has a stronger meaning than real variables is that it has an unlimited approach to holomorphic points compared to real numbers. I may need to add a description of the $$C^n$$ space to make the space we are Integrate more clear. I try read it again without knowing it. Thanks!--SilverMatsu (talk) 13:39, 15 December 2020 (UTC)

Unclear sentence
In the section Radius of convergence of power series, this sentence:

"In the power series $$\sum_{k_1,\dots,k_n=0}^\infty c_{k_1,\dots,k_n}(z_1-a_1)^{k_1}\cdots(z_n-a_n)^{k_n}\ $$, it is possible to define n'' combination of $$r_\nu$$


 * "''$$\begin{cases}

\text{Absolutely converge on}\ \{ z=(z_1, z_2, \dots, z_n) \in {\Complex}^n \mid | z_\nu - a_\nu | < r_\nu, \text{ for all } \nu = 1,\dots,n \}\\ \text{Does not absolutely converge on}\ \{ z=(z_1, z_2, \dots, z_n) \in {\Complex}^n \mid | z_\nu - a_\nu | > r_\nu, \text{ for all } \nu = 1,\dots,n \} \end{cases}$$ ''"

is very poorly worded and makes no sense in normal English. I hope someone knowledgeable about this subject who is also familiar with English can rewrite this so that it is readable and accurate.

I'm guessing that what is meant is this:

... it is possible to define n positive real numbers $$r_\nu$$ such that the power series

$$\begin{cases} \text{is absolutely convergent on}\ \{ z=(z_1, z_2, \dots, z_n) \in {\Complex}^n \mid | z_\nu - a_\nu | < r_\nu, \text{ for all } \nu = 1,\dots,n \}\\ \text{and is not absolutely convergent on}\ \{ z=(z_1, z_2, \dots, z_n) \in {\Complex}^n \mid | z_\nu - a_\nu | > r_\nu, \text{ for all } \nu = 1,\dots,n \} \end{cases}$$

(Is that right?) This would read better if we could get rid of the "cases" curly bracket and just use normal English here.128.120.234.237 (talk) 06:09, 29 December 2020 (UTC)
 * Thank you for the advice. I tried to fix it.--SilverMatsu (talk) 02:04, 31 December 2020 (UTC)

Proposed merge of Several complex variables into Complex analysis
Properly belongs as one article, functions of one complex variable are a special case of functions of several complex variables and both should be discussed in the same article with complex analysis. Furthermore, complex variables is an unacceptable and confusing DAB page which should also redirect here. If those articles need a DAB, it can be accomplished with a hat note and a link to a new complex variables (disambiguation). Size is not an issue. Footlessmouse (talk) 07:03, 31 October 2020 (UTC)
 * Nice to me you. I'm in support of inserting Several complex variables into the complex analysis page, but please wait a bit for page consolidation.In complex analysis, holomorphic is a characteristic property, which is different from several real variables.In other words, we need to write about the differences from several real variables.In order to clarify the difference from complex analysis, it may be possible to integrate several pages included in Category:Several complex variables, or to have duplicate contents.--SilverMatsu (talk) 13:11, 1 November 2020 (UTC)