Talk:Holomorphic function

message: the logarithm is holomorphic on the set {z not a non-negative real number}?  is the correct?  so the logarithm is not holomorphic on positive real numbers?  that can't be right.....?  perhaps it should read {z not a non-positive real number}?  i.e. the logarithm is holomorphic on the positive reals, plus the complexes with nonzero imaginary part?  -Lethe | Talk 22:15, Feb 11, 2004 (UTC)
: I think you're misreading. The log is holomorphic on the set C minus the subset z where z is a negative real. Maybe I'm misreading. The article could be more clear, for sure. 68.103.38.9 12:09, 23 Oct 2004 (UTC)
replies: text: you can choose any axis from the origin to exclude from the domain, much as for root-functions, I think. --MarSch 11:12, 21 October 2005 (UTC)
replies: text: Maybe the page Complex_logarithm#Definition_of_principal_value would clarify. The log is multivalued, and there is no unique way to choose a holomorphic determination of the log (even under the condition that it coincide with the traditional real valued log on the positive real axis). More fundamentally, you can determine homomorphically the multivalued complex logarithm on any simply connected open subset U of C minus 0 (meaning U does not contain any closed path around 0), better yet than having the freedom to choose any axis you can even choose any path that starts at 0 and diverges to infinity (like spirals). Don't know if that helps. 46.193.1.224 (talk) 08:48, 12 March 2017 (UTC)
thread_title: Logarithm question

message: the explaination on the page is a bit to complex for me to understand, can anyone offer a dumbed down version of it suitable for a a-level maths student?
:Most of the article deals with equivalent definitions and properties, but the formal definition does require the notion of complex differentiation, there is simply no going around it. A function is said to be holomorphic on an open set iff complex differentiable at every point of that set. If you want a dumbed down version of complex differentiation, maybe this generalised definition would do the trick.
:"f differentiable at a iff exists df_a linear continuous such that f(x+h)=f(h)+df_a (h)+o(h)"
:complex differentiable means df must be C-linear, complex linear. If real linear means df(lambda x) = lambda df(x) for lambda real, complex linear means in addition df(ix)=i/df(x). From this you deduce all sorts of equivalent definitions, one of them being the Cauchy-Riemann equations. 46.193.1.224 (talk) 08:57, 12 March 2017 (UTC)
replies: text: You wrote "real linear means df(lambda x) = lambda df(x) for lambda real".  That's simple and clear.  The article says instead, "This concept of complex differentiability shares several properties with real differentiability: it is linear...".  That isn't at all clear: What is "the concept", that is linear?  The most-obvious candidate is that it means that f'(x) is linear; but that's obviously false.  I came here only to ask what "this concept is linear" meant.  "df(lambda x) = lambda df(x)" explains that, but isn't in the article (and certainly isn't obvious from the linked-to definition of linear, which says a function which preserves addition and multiplication). Philgoetz (talk) 20:21, 20 August 2023 (UTC)
thread_title: What is a holomorphic function?

message: Could someone explain what this means more precisely? Does "sequence" mean _Cauchy_ "sequence" here? (Also, what is the definition
of a "Cauchy" sequence for complex numbers? Is does it just use the modulus for the "epsilon" in the epsilon-delta argument?).
: It means a sequence zn such that |zn-z0|→0 as n→∞. And about the definition of Cauchy sequences, you are right, you just use the complex modulus instead of the real modulus, everything else is the same as for sequences of real numbers. Oleg Alexandrov 18:30, 1 Mar 2005 (UTC)
replies: text: Yes, this is another way of proceeding about it. Oleg Alexandrov 18:30, 1 Mar 2005 (UTC)

text: Do you still have questions about these? By the way, it is good to sign your comments, with four tildas ~~~~~.  Cheers, Oleg Alexandrov 18:30, 1 Mar 2005 (UTC)

text: I fixed this problem, how do I indicate this? 69.239.185.121 (talk) 04:57, 9 April 2009 (UTC)

text: "why are we defining it by sequences" → defining using neighbourhoods, or metric (for all epsilon exists delta), or sequences are all equivalent definitions of the limit. Think about it like this. Even in real calculus you use the definition of "approaching from all possible paths" except the only two ways you can approach are from the left and from the right. When the two do not coincide, we say the limit is undefined. Similarly, what makes a limit in higher dimension space undefined (in particular, when working in the complex plane C) is that there be two paths (or two sequences) converging to the same point (of study where we want to know if the limit is defined) and for which the limit along each path (or the lim f(zn) as n goes to infinity) does not agree 46.193.1.224 (talk) 09:13, 12 March 2017 (UTC)
thread_title: "The limit here is taken over all sequences of complex numbers approaching z0," ? 

message: does someone know offhand a reference on extending the notion of "holomorphic" or "analytic" to infinite dimensions?  there was a recent edit that claimed frechet differentiability was it, but that can't be right.  (and the text that was there before wasn't right either - the article on frechet differentiability doesn't discuss the notion of a holomorphic function.)  i'm not sure what i wrote it right either... :(  thanks.  Lunch 23:11, 20 June 2006 (UTC)
replies: text: nevermind - i didn't see the discussion above.  btw, it seems the information on banach spaces over the field of complex numbers was removed from the frechet derivative article.  Lunch 23:18, 20 June 2006 (UTC)
replies: text:  See infinite-dimensional holomorphy. That article is not complete, but it gives some idea of what is going on. Oleg Alexandrov (talk) 04:46, 21 June 2006 (UTC)
thread_title: extension to infinite dimensions

message: There's this sentence near the top of the article:
:The phrase "holomorphic at a point a" means not just differentiable at a, but differentiable everywhere within some open disk centered at a in the complex plane.
To me, the sentence sounds like a function can actually be differentiable at a point without being differentiable in some neighbourhood around it. Correct me if I am wrong, but doesn't (complex) differentiability at a point automatically imply that it's differentiable on some neighbourhood around a? -- Amorette 09:50, 17 October 2006 (UTC)
replies: text: No. I think that the function f(x+iy)=x^4 is complex differentiable at zero but not in a neighborhood of zero. Oleg Alexandrov (talk) 15:43, 17 October 2006 (UTC)
thread_title:  Holomorphic at a point 

message: one assertion does not an article make. Aaronbrick 22:19, 10 December 2006 (UTC)
replies: text: The fact that any holomorphic function is analytic is an extremely important assertion. Cramming the proof of that onto this article is not a good idea I think. Oleg Alexandrov (talk) 03:35, 11 December 2006 (UTC)
replies: text:  can you cite any precedent for an article with a sentence for a title? since such functions are the subject of the sentence, i believe it should be documented in their article. Aaronbrick 05:28, 11 December 2006 (UTC)
replies: text: It's not exactly a precedent, but I recently wrote Solving quadratic equations with continued fractions.  ;^>  DavidCBryant 16:06, 2 January 2007 (UTC)

text:  The article holomorphic functions are analytic was moved to Analytic nature of holomorphic functions. I think that name is just vague. I moved it to Proof that holomorphic functions are analytic. That one kind of sucks too, but I can't come up with anything better. Oleg Alexandrov (talk) 17:27, 11 December 2006 (UTC)
thread_title: merge

message: I decided to boldly change the ln(z) in the article to log(z), as oftentimes ln implies that the argument is a real number, which isn't necessarily the case (especially since z is generally used to denote a complex number).  I think that log(z) is better notation, as used in Ward and Churchill's canonical text.67.142.130.18 03:53, 23 February 2007 (UTC)
thread_title:  ln vs log 

message: What's the difference (similarities) between the two?  — Preceding unsigned comment added by Lbertolotti (talk • contribs) 17:38, 7 June 2011 (UTC)
replies: text: To put it simply, morphism are functions, and you add a prefix to mean what condition you put on the morphism. Holomorphism between A and B subsets of C is a holomorphic function from A to B, i.e. a complex differentiable function, such that the inverse too is holomorphic (and is required to be well defined). A diffeomorphism is a real differentiable function such that the inverse is too (and exists). An homeomorphism is just a continuous function, such that the inverse (exists and) is continuous. There are also homomorphisms, functions that commute with an operator (as in f(a)*f(b)=f(a*b) with * denoting some binary operation, or rather two of them, one defined on A and the other defined on B where f is a function from A to B). 46.193.1.224 (talk) 09:24, 12 March 2017 (UTC)
thread_title:  Holomorphism and Homeomorphism 

message: Sorry but you can't say that the definition of analytical function is debatable, then use the term without specifying which definition
you use. Also if wikipedia is sitcking to a particular definition of "analytical function" which is okay, and there is a warning that sometimes in the literature that term is used in a different sense, then that warning should be on the "analytical function" page, not here.  Furthermore the bearing of the paragraph is the relation between holomorphic and analytical functions. The side path totally succeeds in obscuring this.
I don't feel qualified to improve this section, so I just notify.  — Preceding unsigned comment added by 80.100.243.19 (talk) 15:12, 3 February 2013 (UTC)
thread_title:  Analytic functions 

message: The article is currently too technical for non-experts to understand; I am adding a tag to suggest the article be improved to be understandable to non-experts. Betanote4 (talk) 18:15, 5 August 2020 (UTC)
thread_title:  Overly technical 

message: Hi D.Lazard,
Please stop reverting the explanation of the term holomorphic provided by the authors who coined it. I went to look this up after being inspired by a video lecture created by Richard Borcherds wherein he says:
The point of adding a further explanation (drawn directly from the original source) is to see if we can clear up this confusion. If a Fields Medalist can be tricked into thinking that holomorphic is intended to mean "whole shape" based on sources which list (just) the type of etymological information previously included in Wikipedia, then anyone can. But the original intention of the term was something more along the lines of "resembling a whole function" and "resembling a fractional function". That is, the -morphic part was supposed to mean "shaped like". This is substantially less "mysterious". My translation of the original:
–jacobolus (t) 18:03, 12 September 2021 (UTC)
thread_title:  Intended meaning of the term holomorphic 

message: Greetings! In response to this revert, I guess I should explain I came to edit this page because it showed up on a list of articles using HTML entities that have workable alternatives. HTML entities are often disfavored because the vast majority of editors don't know what they are, so they make wikitext more difficult to understand and edit. (And MOS:MARKUP favors wikitext over HTML in general for the same reason.)
In this case,  or  are available and could be used instead, and this article would stop showing up on that report. But there are other considerations which led me to drop hair and thin spaces entirely.
Primarily, MOS:MARKUP also says to "keep markup simple". Fine control over whitespace might be one of the things we don't need to do if the result of simple markup is acceptable for most readers. Results do vary across operating systems, web browsers, and installed fonts, so optimizing appearance for one editor may have undesirable results for some readers. I think there's a reasonable argument for sticking in a hair space between characters that actually overlap, which sometimes happens when the first of a pair is italicized and the second isn't. (Though if the rendering system causing this problem gets fixed, that could suddenly result in unwantedly large amounts of space as a side effect of the fix.) But the difference between something like  and  doesn't seem worth the extra complexity. The former is actually closer to what I would write by hand, so I'm not sure the version with more space is prettier, and it's not clearer.
The general practice for "simple" non-LaTeX markup for equations, including on MOS:MATH, seems to be either to use full spaces or none. That approach makes the equation almost WYSIWYG because there aren't strings of letters and punctuation interjected where normally there would only be whitespace or no space.
Given all that, do you think most or all of the hair and thin spaces in this article could be dropped? If beauty is a strong motivating factor and you find it would be more visually attractive, conversion to  markup is also a perfectly reasonable solution that would seem to align with general practice. I guess math editors are just de facto expected to know or be able to learn that syntax.
Oh, and I'll also note you reverted conversion of a curly apostrophe; straight apostrophes and quote marks are required by MOS:STRAIGHT. -- Beland (talk) 01:34, 6 February 2022 (UTC)
replies: text: Feel free to switch the apostrophe to a ' glyph, following prevailing Wikipedia anti-typography. –jacobolus (t) 05:01, 6 February 2022 (UTC)
thread_title:  Whitespace and punctuation