Talk:Function of several complex variables

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)

@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!--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)

I'm considering merge the domain of holomorphy into Several complex variables, but the domain of holomorphy (theory) may be the title of this page

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 EOM makes me think about this. The title of the textbook uses several complex variables, so my idea may be off the mark. thanks!--SilverMatsu (talk) 05:33, 21 December 2020 (UTC)

I forgot to say it. Redirecting to the title of this page instead of section redirecting to the domain of holomorphy is also a suggestion choice.--SilverMatsu (talk) 05:39, 21 December 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 $\mathbb {C} ^{n}$ or on Stein manifolds, which is what the current lead gives most of its weight to.

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! Tazerenix (talk) 05:53, 23 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!--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!--SilverMatsu (talk) 13:53, 26 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 $r_{\nu }$

"${\begin{cases}{\text{Absolutely converge on}}\ \{z=(z_{1},z_{2},\dots ,z_{n})\in {\mathbb {C} }^{n}\mid |z_{\nu }-a_{\nu }|<r_{\nu },{\text{ for all }}\nu =1,\dots ,n\}\\{\text{Does not absolutely converge on}}\ \{z=(z_{1},z_{2},\dots ,z_{n})\in {\mathbb {C} }^{n}\mid |z_{\nu }-a_{\nu }|>r_{\nu },{\text{ for all }}\nu =1,\dots ,n\}\end{cases}}$

"

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

${\begin{cases}{\text{is absolutely convergent on}}\ \{z=(z_{1},z_{2},\dots ,z_{n})\in {\mathbb {C} }^{n}\mid |z_{\nu }-a_{\nu }|<r_{\nu },{\text{ for all }}\nu =1,\dots ,n\}\\{\text{and is not absolutely convergent on}}\ \{z=(z_{1},z_{2},\dots ,z_{n})\in {\mathbb {C} }^{n}\mid |z_{\nu }-a_{\nu }|>r_{\nu },{\text{ for all }}\nu =1,\dots ,n\}\end{cases}}$

(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)