Talk:Function of several complex variables/Archive 1: Difference between revisions

Revision as of 13:59, 26 December 2020

Latest revision as of 09:29, 19 April 2021

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{{archive}}

== Vote for new external link ==

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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)

== 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.--[[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)

== 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. --[[User:SilverMatsu|SilverMatsu]] ([[User talk: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. --[[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)

:::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. [https://www.jstor.org/stable/2323391 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. [[User:Footlessmouse|Footlessmouse]] ([[User talk: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]].--[[User:SilverMatsu|SilverMatsu]] ([[User talk: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!--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 06:20, 24 December 2020 (UTC)

:Addendum:Therefore, templates seem to be better for functions than complex analysis.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 06:22, 24 December 2020 (UTC)

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

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> on the Reinhardt ___domain D, including the center a, has been shown to converge uniformly on any compact set on D<ref group=Ifaptmbrttp>H. Cartan, ''Les fonctions de deux variables complexes et le problème de la représentation analytique'' J.de Math.(9),10,1931,p.19</ref> 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)

::::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". [[User:Footlessmouse|Footlessmouse]] ([[User talk:Footlessmouse|talk]]) 11:19, 12 December 2020 (UTC)

:::::Thank you for your advice. I will add it to the See also.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 11:44, 12 December 2020 (UTC)

:::::Looking at the [[Compact space#example|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.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 12:20, 12 December 2020 (UTC)

{{Outdent|5}} I changed the reference link of the infinite-dimension page, so it should be available for download. Thanks to [[User:Michael D. Turnbull|Mike Turnbull]] advice.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 13:41, 14 December 2020 (UTC)

I was able to find out the weak holomorphic.

Weak definition <ref group=Ifaptmbrttp>Lawrence A. Harris, ''[https://www.ms.uky.edu/~larry/paper.dir/korea.ps Fixed Point Theorems for Infinite Dimensional Holomorphic Functions]'' (undated).</ref>

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

Since it says ''[[Infinite-dimensional holomorphy#Vector-valued holomorphic functions defined in the complex plane|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 <math>C^n</math> space to make the space we are Integrate more clear. I try read it again without knowing it. Thanks!--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 13:39, 15 December 2020 (UTC)

=== References ===

{{reflist|group=Ifaptmbrttp}}

== Unclear sentence ==

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

"''In the power series <math>\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}\ </math>, it is possible to define ''n'' combination of <math>r_\nu</math><ref group=note>This combination may not be unique.</ref>

:"''<math>\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}</math>

''"

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 <math>r_\nu</math> such that the power series

<math>\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}</math>

(Is that right?) This would read better if we could get rid of the "cases" curly bracket and just use normal English here.[[Special:Contributions/128.120.234.237|128.120.234.237]] ([[User talk:128.120.234.237|talk]]) 06:09, 29 December 2020 (UTC)

:Thank you for the advice. I tried to fix it.--[[User:SilverMatsu|SilverMatsu]] ([[User talk: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. [[User:Footlessmouse|Footlessmouse]] ([[User talk:Footlessmouse|talk]]) 07:03, 31 October 2020 (UTC)

:{{Ping|Footlessmouse}} 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.--[[User:SilverMatsu|SilverMatsu]] ([[User talk:SilverMatsu|talk]]) 13:11, 1 November 2020 (UTC)