Talk:Holomorphic function

426:"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(z 92: 82: 64: 31: 2891:

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).

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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

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Sentences such as the example occur often in literature outside wikipedia and are likely to arise in here often, more often if the hatnote is missing, so those editors who know something about fungi and search for "holomorphic" find nothing relevant for them to link to, then others who think themselves helpful come and add the link to this page here. Removing the hatnote is a recipe for a long-term mess.

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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).

2194:. And all of my texts for complex analysis use the term analytic function to refer to a holomorphic function. So the unfortunate fact is that this will cause confusion. Also from the standpoint of math, it would be a problem to have two separate article for the same thing. For example, how do you decide which article some material you want to add should go to. -- 3167: 3127: 206:

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.

3297:, 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 2070:

defined and not in substance (or maybe I am wrong). At least, if we wikipedia claim that they are two different things we should give an example of both functions, one of which that is analytic but not holomorphic and one that is holomorphic but not is analytic. I don't know if there exists such functions. --

1459:. Holomorphic functions, from what I know are differentiable functions of complex argument. To define the derivative in respect to a variable in the Banach space, you should be able to divide things (the denominator of the derivative). Things could be messy. We indeed need references before moving on. 3034:

The answer to the question about who hatnotes are for, is both. Knowledge is a community of users who also contribute so the distinction is a false dichotomy. Obviously I agree with the latter point that people who create links should check them before saving, but that was not the central issue which

2911:

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

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Thanks James, that explained it. I had not grasped the various possibilities for a function having a Taylor series yet that series not converging to the right function. It's all straightforward with that resolved. Feel free to delete this discussion if you think it clutters the Talk. Thanks again.

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would be wrong. They talk about different things, using different math (power series for analytic funcions, path integration for holomorphic functions). The reader would learn best if he/she would understand that analytic and holomorphic are different concepts, which sometimes coincide, but only when

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Holomorphic functions are complex functions which are differentiable. That a holomorphic function is complex analytic is a very profound theorem using intimate properties of the complex number field, and it has dramatic consequences (like every polynomial with complex entries has complex roots). Yes,

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Thanks for the replies, and Quondum I'm very happy to see that you are convinced. About the preponderance of the mathematical meaning wikipedia: I'd argue that that is partly historical, and should not be taken as an accurate measure of the rarity of the meaning related to fungi. That falls into the

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The predominance of the mathematical use is huge in WP, so I do not feel intensely about this. Nevertheless, the redirect pointing at the redirect is a good solution, because I've relinked all occurrences of these links throughout WP (en), and new dablink occurrences will automatically be flagged by

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Do you really think that hatnotes are for editors rather than for readers? IMO, if incorrect links are being created by editors, the place to fix that is at the link. Besides, an editor that links to an article that is so clearly unrelated to the subject matter is clearly not looking at what he/she

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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

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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?

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is wrong. Supposing you are right, then I don't think there is a point to have separate articles for the essentially same kind of functions. I understand analytic function has a meaning in real analysis. But I highly doubt people mean a real function when they use a term analytic function. What I am

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I agree with you that this note would be too technical, and many people complain about articles beeing to technical. However, you could add a section, preferrably somewhere to the bottom, entitled ==Connection with complex geometry==, where you can put that observation. That is, such text is good to

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You probably can't define "holomorphic" on a space without a complex structure. When your space is C^n, the complex structure is obvious (it multiplies each "z" by i, and each z* by -i), but when you say Banach space, there is no canonical complex structure. what are your holomorphic coordinates?

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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

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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,

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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

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in that part of the plane. We mean by this name that it resembles entire functions which enjoy these properties in the full extent of the plane. ¶ A rational fraction admits as poles the roots of the denominator; it is a holomorphic function in all that part of the plane which does not contain any

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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 Knowledge, then anyone can. But the original intention of the term was something more along the lines of "resembling a whole

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The structure of roots is very simple, roots cannot get clustered together, unless we talk about the zero function. This is already in the article. I know too little about analytic functions with values in Banach spaces or linear topological spaces to write about it, but I doubt that anything truly

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I remember reading an article which defined what holomorphic was, but it doesn't seem to be this on (any more?). Anyway it is a theorem that a holomorphic function is a complex analytic function and vice versa and the article reflects this. However analytic might also mean real analytic, so I guess

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Thank you, Oleg and Linas, for your thoughts. I will see what is out there and try to write something both useful and not too redundant with existing material. As should be clear, I am new to this : do I post my proposed section here for discussion and approval, or do I go ahead and edit the real

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The biggest problem with WP is not lack of content but lack of organized content. There are plenty of advanced articles and plenty of beginner articles, but many holes in between. Things aren't linked that should be. Sometimes articles discuss a well known topic under a weird title, and so someone

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in an article about a fungus species, for example in a sentence "The genus as circumscribed by Bateson was holomorphic." and ends up here, the most likely next action of the reader is to despair of wikipedia. Hatnotes are used to guide the reader, that is their only function and it is a noble one.

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Ah, I see what the problem is. I was stuck at assuming that a function either has a Taylor series representation or it doesn't. Apparently a function can have a Taylor series representation that fits only part of the domain of the function, and at other parts of the domain it does not fit. OK, so

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I guess I was thinking of taking the difference betwen real differentiability on R^2 (derivatives may vary according to direction) versus complex differentiability on C^1 (derivative is independent of direction of variation), and extending that to higher dimensional spaces. But I guess that's not

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The issue is that if you search for holomorphic you end up here, which is fine for a mathematician but not so good if you are interested in fungi. The point is that redirects do just that, redirect or rescue people, and help them find what they are looking for. The alternative is a disambiguation

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Oleg, I am not suggesting we shouldn't have an article about analytic functions that are not complex. (so they may not be holomorphic). I certainly never claimed that there is no such thing as real analytic function. My problem is that an analytic function is a very common term used to refer to a

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This seems like a strange way to define differentiability to me. lim already has a meaning in the complex numbers, so why are we defining it by sequences? Also my intuition is that if we want to do it this way we should be talking about nets not just sequences. I'm going to think about this for a

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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

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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

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People very much use the term "analytic function" for real analytic functions. An analytic function is a function which is locally expandable in power series. You don't need absolutely no complex analysis to discuss about these functions. That is why I think it is good to have a separate article

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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

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synonymous with "holomorphic function of one complex variable". An analytic function can be real analytic, complex analytic, analytic with coefficients in a Banach space or in a linear topological space, and so on. It can be of one variable, of several variables, and I think even of an infinite

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I think I am being ignorant but I really cannot understand the difference between holomorphic functions and analytic functions. Texts I have say they are the same thing. This article is the only one I know that claims otherwise. I believe the difference is how holomorphic-ness or analyticity is

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which, not to put too fine a point on it, has been a disaster area here that anyone with any knowledge of the subject would just despair of trying to fix. I hope that can change in time, and then the educational power of wikipedia might cause a small snowball effect to bring more mentions of

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I daresay WP covers topics more technical than that; that's not a problem. If what you want to write is longer than a section, you can start a new article. What you are asking for is already partly discussed, piece-meal, in some scattered articles, all over WP, with no particularly coherent

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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

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A holomorphic map is one whose derivative with respect to the antiholomorphic coordinates vanishes, but without choosing a complex structure, how can you choose which coordinates those are? So far, it seems like you're only talking about analyticity, which isn't quite the same thing. -

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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.

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still complex, but a_n in the Banach space. I would not want us to get into that, unless such an article/section will have some interesting phenomena, that is, beyond a dry generalization. So I would agree with you. By the way, such a thing is useful in proving that the

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You go ahead and edit the real page directly. If you encounter resistance then you can go to talk to discuss and solve any problems. When you change substantially what is already in an article it is usually a good idea to mention it on talk first and gauge reactions.

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Well, whatever the case may be, the use appears to be established. The thing I'm reading now uses the Earle-Hamilton theorem to find a fixed point which is identified with the equilibrium state in statistical mechanics. (In my case, for the equilibrium state for the

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When the words "equal to its own Taylor series" occur it has already been stated that the function is infinitely differentiable, so it is certainly possible to formally write down a Taylor series for the function. Now it is perfectly easy to have a real function

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And about the references you mention (MathWorld, PlanetMath). In the first one only the case of complex analytic functions is discussed, when again, that is the same as holomorphic. And PlanetMath has separate articles for analytic and holomorphic, same as us.

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Yep. OK, I have a reference to a paper that would have to define the term, as it states theorems about it. I'll look it up. And actually, what I was reading did say "Banach space" and not "Banach algebra". Though, in context, the things discussed were algebras.

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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.

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are just a bunch of constants. So it seems that the statement "its a holomorphic map on a Banach algebra" just means that "the map can be written as a power series". Well, OK. Dohhh. I guess that means that, naively, a "holomorphic map" can be defined for any

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for complex numbers c_n. Division is not a problem, a Banach space is a vector space, and so differentiation is just like on a vector space -- I can choose to go in any direction by a small amount; the denominator is just a complex number. Time to go to bed.

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entry linked above. At the end of the section it says: "A hatnote may be appropriate in an unambiguously named article when an ambiguous term redirects to it, as explained in the "Proper uses" section above." "Holomorphic" is an ambiguous term that redirects

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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?).

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I'm pretty sure what this sentence is trying to say is just that holomorphic functions can be represented by a Taylor series. It might even be trying to say that if a function is representable by a Taylor series, then it's holomorphic.

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I hope that this has clarified things, but this discussion has moved away from improving the article onto a general discussion of the mathematics involved, so I suggest that this is not the place to pursue the issue any further.

3055:, which pointed here. I hadn't realized that. Given that the term has diverse meanings in different disciplines, it seems to me that that should be replaced with a disambiguation page, which I see exists already in the guise of 2256:

Fine. Expanding analytic functions should even make it clear why we have the separate articles. Also, my question of where to put, that was little too hypothetical. If we really had this problem, we can discuss again then. --

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Err, no, not at all. At least, well, I'm pretty sure that is not what was meant. I was confidently reading something that was talking about holomorphic maps on Banach spaces, and I was assuming that (for finite N) this meant

3353:, 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. 2216:

Complex analysis is just a small part of what analytic funcions are about. To answer your question about what to put where. Material about analytic functions (power series expansions, structure of roots, etc) should go to

2311:. But again, analytic and holomorhic are different concepts, and they should be in different articles. And I said it before, don't take for granted that any holomorphic function is analytic. It is a miracle of sorts. :) 1086: 3059:, so I have changed the redirect to point to that page. Perhaps my arguments could be reevaluated given this change. I trust the mathematics community will not be so possessive of the term as to disagree with this. — 2286:

Is this a tricky question? You seem to know the answer; you have given a plenty of possibilities. structure of roots, topology that has an analytic function, or the case when coefficients in a Banach space, etc. --

3131:– doesn't it always?) Yeah, this is definitely not black-and-white. I guess that as long as hatnotes don't proliferate unduly, it will be about a balance of preferences. Notice that since I discovered the redirect 1445:

Seeing as I tripped on this, I suppose a statement along these lines could be added to some article somewhere ... is there an authoritative reference that could be checked, just so that I'm not imagining things?

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in a book of complex analysis people will use the term "analytic function" to refer to "complex analytic function". But this is an encyclopedia, not a book a complex analysis. The words "analytic function" are

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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

771:). At least, thats what I assumed. So I thought I'd look it up here. I'm now confused. Unfortunately, it wasn't a textbook, so it didn't explain the term before using it. I'll see if I can unconfuse myself. 655: 1433: 1936: 1834: 709: 1549: 365:|→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. 1364: 900: 2019:

However, overzealous organization can do more harm than good. :) And some repetition and overlaps are OK, it never hurts to see the same thing from different perspectives/different context.

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DPLbot (and hopefully in response will be disambiguated by the people who create them when thus notified), both in math and biology articles. The alternative, with mistaken linking to

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the sentence in question here is actually saying "equal to its Taylor series over the entire domain" or "has no part of the domain where the Taylor series is inapplicable" or somesuch?

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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.

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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.

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means 'whole shape' or 'whole form'. And I have no idea why this was used to mean a differentiable complex function. Um... Maybe someone just liked obscure Greek words or something.

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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.

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in biology articles generally going unrecognized, despite being infrequent, just to save the effort by math editors of fixing dablinks (while creating the need for the hatnote that

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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.

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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.

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Hmm, the definition given in that paper is identical to what I know as the normal Fréchet derivative in any Banach space. It's the same definition found here on wikipedia in

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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.

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The answer may be googleable. Searched "Earle Hamilton Banach" and I get lots of hits to the "Earle-Hamilton fixed-point theorem for holomorphic maps on Banach spaces".

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are not equal. What the sentence referred to in the article is saying is that for a holomorphic function on the complex plane this cannot happen: that the function

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has linked to, and will not see the hatnote. So you are really saying that it is worth having a hatnote to help a reader understand that the link was misleading? —

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it. I have to say, though, I don't like this usage of the word "holomorphic". The Fréchet derivative depends on the direction of variation of your derivative. -

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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.

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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.

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Either way... if someone could clarify this... it would be much appreciated (I am not a Math expert - stumbled across this trying to learn some calculus stuff).

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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.

2533: 454:, that is, where both the domain and codomain is some subset of a Banach space. This seems straightforward for finite dimansional Banach spaces (by analogy to 2793:) is like this: it converges to the function which is identically zero. However, in the complex case this cannot happen: where the Taylor series converges it 35: 2622:

will always be equal to the function defined by its Taylor series. (A simple example of a real function which has a Taylor series but is not equal to it is

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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?

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I suppose the other way to define all this is by talking about the real and imaginary parts each converging to the real and imaginary parts of z0...

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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

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Oleg, thank you, yes, that was exactly it; I meant Banach algebra and not Banach space, and that was the source of my confusion. So I guess that

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functions it is possible to have a function for which we can write down a Taylor series, and that Taylor series converges throughout the domain,

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When a function is continuous, monotropic, and has a derivative, when the variable moves in a certain part of the plane, we say that it is

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list, the only mainspace links that I could find were mathematical topics or disambiguation pages. Other opinions would be appreciated. —

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Is this suggesting that some other kind of function could have a Taylor series representation and NOT be equal to it? This is confusing.

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eventually complex analytic functions and holomorphic functions turn out to be the same thing, but this should not be taken for granted.

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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

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is that that there are two pages, and the search term points here, and would not answer the question of what does holomorphic mean? --

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I now know more about Banach spaces than I used to. The definition of holomorphic functions on Banach spaces is given in the article

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differentiability doesn't discuss the notion of a holomorphic function.) i'm not sure what i wrote it right either... :( thanks.

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starts a new article under the well-known name, repeating the content. Surf around a bit. See what you can find, fill in the holes.

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So holomorphic is by definition (1 times) complex differentiable? Is holomorphic ever used for other then complex functions? --

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Oh, and I'll also note you reverted conversion of a curly apostrophe; straight apostrophes and quote marks are required by

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The downside is that this rapidly becomes very technical, and perhaps beyond the scope of an encyclopedia. Any thoughts?

1481: 2225:. I think you never encountered analytic functions outside complex analysis, and that might explain your point of view. 389:

Do you still have questions about these? By the way, it is good to sign your comments, with four tildas ~~~~~. Cheers,

44: 1292: 862: 2677:, with f(0)=0. It is easy to show that all derivatives at the origin are zero, so the Taylor series is also zero.) 483:, I guess), but additional insights, theorems, references, etc. for infinite-dimensional spaces would be welcome. 3252:

part was supposed to mean "shaped like". This is substantially less "mysterious". My translation of the original:

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I had no idea this could get so complicated - but are hats all that bad? I mean they do have an educational role--

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was a one-(complex)-parameter family of maps, and I'm pretty sure that is not what was meant. Rather, the map

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Is the function of a hatnote *really* to help a reader discover that the link they've just followed is a dud?

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want a dumbed down version of complex differentiation, maybe this generalised definition would do the trick.

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As always these disagreements can lead to useful discussions. The note that was left on my talk page reads:

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you can choose any axis from the origin to exclude from the domain, much as for root-functions, I think. --

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Never heard that word before in my life. I'm suspicious. Maybe it's a nonstandard term for branch cut? -

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poles. ¶ When a function is holomorphic in part of the plane, except at certain poles, we say that it is

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page. I suspect that most people interested in either mathematics or fungi are unaware of the ambiguity.

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The inverse trigonometric functions likewise have seams and are holomorphic everywhere except the seams.

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necessarily the case that it converges throughout the domain of the function, as in the case of 1/(1-

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holomorphic function. Maybe people are wrong in doing this, but isn't that the reality? For example,

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Feel free to switch the apostrophe to a ' glyph, following prevailing Knowledge anti-typography. –

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In the intro, what does this mean: "any holomorphic function is equal to its own Taylor series."

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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I added a hat when I discovered that the use of holomorphic in articles on fungi linked here. --

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holomorphic fungal taxonomic treatments (as we in the botanical trade relish calling them).

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which has a formal Taylor series, and for which that Taylor series converges to a function

3318: 3308: 2499: 2475: 2307:, the way it is now, does not provide too much justification for keeping it separate from 2001: 1946: 714: 3385: 3288: 3270: 3190: 3110: 3094: 3036: 2988: 2976: 2957: 2956:... which I reverted, on the grounds that second-guessing blatantly incorrect links is 2942: 1776: 1740: 1732: 1703: 1627: 1567: 1028: 300: 168: 1975:

have in a standalone section if some people would like to skip it. What do you think?

3400: 2189: 1666: 2192: 3366: 3244: 2718:

No, that is not the point. For example the Taylor series about the origin for 1/(1-

1691: 1662: 1617: 451: 1604: 3072:. I've made another try by reverting that change. Fungi forever! (or something). 2184:

MarSch, at least the current article says that a holomorphic function is complex.

1661:

Anyway, I'm planning on moving discussions about differentiability from both the

344:"The limit here is taken over all sequences of complex numbers approaching z0," ? 3370: 3069: 2980: 2586:

If one or both of these is correct, perhaps the intro could make this clearer?

2170: 2082: 2053: 1643: 191: 110: 3391: 3378: 3349:

The general practice for "simple" non-LaTeX markup for equations, including on

3276: 3198: 3179: 3141: 3118: 3104: 3081: 3063: 3044: 3024: 3001: 2970: 2950: 2930: 2900: 2882: 2845: 2818: 2705: 2686: 2595: 2565: 2545: 2517: 2502: 2493: 2478: 2460: 2394: 2368: 2346: 2336: 2320: 2291: 2280: 2261: 2250: 2234: 2198: 2173: 2156: 2109: 2085: 2074: 2056: 2042: 2028: 2014: 1984: 1968: 1770: 1677: 1650: 1610: 1598: 1589: 1560: 1463: 1450: 1035: 979: 775: 552: 487: 439: 418: 402: 270: 253: 215: 194: 3328: 3298: 3212: 2343: 2333: 2011: 1767: 1674: 1647: 1607: 1595: 1586: 1557: 1447: 976: 772: 484: 87: 3056: 2359:. That article is not complete, but it gives some idea of what is going on. 2101:

suggesting is that to merge analytic function here and create a new article

450:

If I'm not mistaken, one can define the concept of holomorphic functions on

3265:

in that part of the plane, that is to say it resembles rational fractions.

3350: 1476:(I slap my forehead). Indeed. I guess I missed the "obvious" definition: 2536:. That one kind of sucks too, but I can't come up with anything better. 1836:(as currently defined) is also equivalent to saying that if we identify 1027:

For that, you need multiplication on B, don't you? So, you would need a

1800:

It might be useful to mention that complex differentiability of a map

1690:

I'm all for that. But of course, the derivative that is currently in

2451:

is complex differentiable at zero but not in a neighborhood of zero.

1739:

to that effect though, so it'll be clear after you make the move. -

2915:

I don't feel qualified to improve this section, so I just notify.

331:

It is a small insignifcant fragment anyway. So what do you think?

1887:, the derivative Df(p) is complex linear (i.e. commutes with the 1731:(for reasons I'm not perfectly clear on, maybe the TVS has to be 3248:

function" and "resembling a fractional function". That is, the

3068:

Apparently so (the possessiveness), since you were reverted at

2960:. My revert was then reverted. Searching through the lengthy 2105:

or something. That reflects the real uses, to my knowledge. --

2726:| < 0, although the function is holomorphic on the domain { 1992:

presentation. For example, I remember seeing a discussion of

1728: 650:{\displaystyle \sum _{k=0}^{N}a_{k}f_{k}(z_{1},\ldots ,z_{N})} 15: 3301:

favors wikitext over HTML in general for the same reason.)

3049:

I see that the problem may be rooted in the redirect page

1428:{\displaystyle c_{k_{1}k_{2}\ldots k_{N}}\in \mathbb {C} } 2271:

to make it clear that it is not the same as holomorphic?

1455:

I would not be so quick. What you have so far is a fancy

1073:

is holomorphic if it can be written as a power series in

3294: 3133: 3097: 3089: 3051: 3017: 2862:

What's the difference (similarities) between the two?

1931:{\displaystyle T_{p}\mathbb {C} \cong \mathbb {R} ^{2}} 1829:{\displaystyle f:\mathbb {C} \rightarrow \mathbb {C} } 704:{\displaystyle f_{k}:\mathbb {C} ^{N}\to \mathbb {C} } 2744: 2628: 2406: 2221:. Material specific to complex analysis should go in 1897: 1864: 1842: 1806: 1484: 1376: 1295: 1089: 1047: 997: 949: 908: 865: 794: 744: 717: 667: 570: 501: 460: 2511:

Solving quadratic equations with continued fractions

1544:{\displaystyle f(a)=\sum _{n=0}^{\infty }c_{n}a^{n}} 109:, a collaborative effort to improve the coverage of 2509:It's not exactly a precedent, but I recently wrote 2381:There's this sentence near the top of the article: 2785: 2669: 2572:I suggest clarifying relationship to Taylor series 2443: 1930: 1879: 1850: 1828: 1543: 1427: 1359:{\displaystyle a=(a_{0},a_{1},\ldots ,a_{N})\in B} 1358: 1275: 1065: 1015: 967: 935: 895:{\displaystyle g_{k}:\mathbb {C} \to \mathbb {C} } 894: 847: 763: 730: 703: 649: 527: 475: 167:plus the complexes with nonzero imaginary part? - 143:This article has not yet received a rating on the 2532:. I think that name is just vague. I moved it to 377:Yes, this is another way of proceeding about it. 289:Can some expert clarify what a seam is? Thanks! 3427:Knowledge level-4 vital articles in Mathematics 3254: 3221: 1031:to replace the complex number. Messy business. 2267:Any concrete suggestions about what to put in 1961:(as introduced by Gromov in '85, Inv. Math.). 409:I fixed this problem, how do I indicate this? 2534:Proof that holomorphic functions are analytic 2081:this is what the article is trying to say. -- 8: 3137:, I've not been arguing about the hatnote. — 848:{\displaystyle \sum _{k=0}^{N}a_{k}g_{k}(z)} 1702:, so it needs to be moved there instead. - 1941:This then makes clear the connection with 1723:Also, just for clarity, the derivative in 902:was holomorphic. But that would mean that 58: 2772: 2764: 2743: 2656: 2648: 2627: 2435: 2405: 1922: 1918: 1917: 1909: 1908: 1902: 1896: 1871: 1867: 1866: 1863: 1844: 1843: 1841: 1822: 1821: 1814: 1813: 1805: 1535: 1525: 1515: 1504: 1483: 1421: 1420: 1409: 1396: 1386: 1381: 1375: 1341: 1322: 1309: 1294: 1265: 1260: 1255: 1240: 1235: 1230: 1218: 1213: 1208: 1196: 1183: 1173: 1168: 1158: 1145: 1140: 1127: 1114: 1109: 1088: 1046: 996: 948: 907: 888: 887: 880: 879: 870: 864: 830: 820: 810: 799: 793: 749: 743: 722: 716: 697: 696: 687: 683: 682: 672: 666: 638: 619: 606: 596: 586: 575: 569: 519: 509: 500: 467: 463: 462: 459: 3204:Intended meaning of the term holomorphic 2530:Analytic nature of holomorphic functions 2474:one assertion does not an article make. 2303:As such, you are right that the article 430:) as n goes to infinity) does not agree 3417:Knowledge vital articles in Mathematics 1766:. Review for clarity and correctness. 318:My motto is: When in doubt, cut it out. 60: 19: 2736:but it converges to the wrong function 2722:) converges to the function only for | 446:Holomorphic functions on Banach spaces 3442:Unknown-priority mathematics articles 3432:C-Class vital articles in Mathematics 2795:must converge to the correct function 7: 3164: 3124: 103:This article is within the scope of 3358: 1796:(real) derivative is complex linear 49:It is of interest to the following 2526:holomorphic functions are analytic 1516: 1159: 1128: 14: 1646:.) So much math, so little time. 975:was itself termed "holomorphic". 123:Knowledge:WikiProject Mathematics 3412:Knowledge level-4 vital articles 3165: 3125: 2786:{\displaystyle f(x)=e^{-x^{-2}}} 2670:{\displaystyle f(x)=e^{-x^{-2}}} 2327:extension to infinite dimensions 1880:{\displaystyle \mathbb {R} ^{2}} 1023:was itself termed "holomorphic". 476:{\displaystyle \mathbb {C} ^{n}} 126:Template:WikiProject Mathematics 90: 80: 62: 29: 20: 3227:is a little bit mysterious. So 2979:that if a reader who clicks on 2357:infinite-dimensional holomorphy 2092:No, I am not suggesting either 1735:). I made an edit recently to 528:{\displaystyle \sum a_{n}z^{n}} 224:What is a holomorphic function? 3422:C-Class level-4 vital articles 3277:18:03, 12 September 2021 (UTC) 3231:means 'whole' or 'entire' and 2858:Holomorphism and Homeomorphism 2754: 2748: 2638: 2632: 2425: 2410: 2400:No. I think that the function 2147:the right conditions are met. 2138:As such, merging the articles 1818: 1783:05:29, 25 September 2005 (UTC) 1771:03:40, 25 September 2005 (UTC) 1747:14:44, 22 September 2005 (UTC) 1710:14:29, 22 September 2005 (UTC) 1678:13:38, 22 September 2005 (UTC) 1651:03:09, 22 September 2005 (UTC) 1634:17:54, 21 September 2005 (UTC) 1611:14:13, 21 September 2005 (UTC) 1599:14:10, 21 September 2005 (UTC) 1590:13:51, 21 September 2005 (UTC) 1574:11:24, 21 September 2005 (UTC) 1561:05:26, 21 September 2005 (UTC) 1494: 1488: 1464:04:54, 21 September 2005 (UTC) 1451:04:47, 21 September 2005 (UTC) 1347: 1302: 1099: 1093: 1057: 1036:02:00, 21 September 2005 (UTC) 1007: 980:01:05, 21 September 2005 (UTC) 959: 927: 918: 912: 884: 842: 836: 776:00:57, 21 September 2005 (UTC) 693: 644: 612: 553:04:22, 20 September 2005 (UTC) 488:03:49, 20 September 2005 (UTC) 1: 3180:17:06, 13 December 2014 (UTC) 3142:02:40, 12 December 2014 (UTC) 3119:01:44, 12 December 2014 (UTC) 3105:23:39, 11 December 2014 (UTC) 3082:23:04, 11 December 2014 (UTC) 2846:16:18, 16 December 2009 (UTC) 2819:13:16, 16 December 2009 (UTC) 2706:21:49, 14 December 2009 (UTC) 2687:14:35, 14 December 2009 (UTC) 2596:02:20, 10 December 2009 (UTC) 2566:03:53, 23 February 2007 (UTC) 2546:17:27, 11 December 2006 (UTC) 2503:05:28, 11 December 2006 (UTC) 2494:03:35, 11 December 2006 (UTC) 2479:22:19, 10 December 2006 (UTC) 2444:{\displaystyle f(x+iy)=x^{4}} 1698:, which is distinct from the 403:03:58, 15 November 2006 (UTC) 117:and see a list of open tasks. 3437:C-Class mathematics articles 3392:05:01, 6 February 2022 (UTC) 3379:01:34, 6 February 2022 (UTC) 3235:means 'shape' or 'form'. So 3064:00:45, 8 December 2014 (UTC) 3045:21:52, 7 December 2014 (UTC) 3025:16:36, 7 December 2014 (UTC) 3002:19:45, 7 December 2014 (UTC) 2971:17:27, 7 December 2014 (UTC) 2958:not the purpose of such hats 2951:16:28, 7 December 2014 (UTC) 2931:15:12, 3 February 2013 (UTC) 2738:. The example I gave above ( 2730:≠ 1}. The point is that for 2461:15:43, 17 October 2006 (UTC) 2395:09:50, 17 October 2006 (UTC) 2321:03:12, 25 October 2005 (UTC) 2292:23:37, 24 October 2005 (UTC) 2281:09:35, 23 October 2005 (UTC) 2262:23:27, 22 October 2005 (UTC) 2251:03:12, 22 October 2005 (UTC) 2235:02:57, 22 October 2005 (UTC) 2199:23:15, 21 October 2005 (UTC) 2174:16:47, 21 October 2005 (UTC) 2157:14:26, 21 October 2005 (UTC) 2110:13:31, 21 October 2005 (UTC) 2086:11:20, 21 October 2005 (UTC) 2075:10:49, 21 October 2005 (UTC) 2057:12:39, 8 November 2005 (UTC) 2043:01:14, 8 November 2005 (UTC) 2029:01:21, 19 October 2005 (UTC) 2015:23:59, 18 October 2005 (UTC) 1985:23:54, 18 October 2005 (UTC) 1969:20:20, 18 October 2005 (UTC) 1945:(e.g. holomorphic maps on a 1851:{\displaystyle \mathbb {C} } 786:The other possibility being 493:You mean series of the form 195:11:12, 21 October 2005 (UTC) 2797:. On the other hand, it is 2518:16:06, 2 January 2007 (UTC) 1669:articles to the article on 936:{\displaystyle g(z):B\to B} 271:20:21, 20 August 2023 (UTC) 3458: 3293:Greetings! In response to 3282:Whitespace and punctuation 3199:18:15, 5 August 2020 (UTC) 2901:09:24, 12 March 2017 (UTC) 764:{\displaystyle a_{k}\in B} 440:09:13, 12 March 2017 (UTC) 254:08:57, 12 March 2017 (UTC) 216:08:48, 12 March 2017 (UTC) 175:22:15, Feb 11, 2004 (UTC) 2369:04:46, 21 June 2006 (UTC) 2347:23:18, 20 June 2006 (UTC) 2337:23:11, 20 June 2006 (UTC) 419:04:57, 9 April 2009 (UTC) 307:09:07, Feb 11, 2005 (UTC) 293:06:10, 11 Feb 2005 (UTC) 142: 75: 57: 3223:The reason for the name 2987:Please also re-read the 2975:I completely agree with 2883:17:38, 7 June 2011 (UTC) 2298:unique would take place. 1370:is N-dimensional, and 1066:{\displaystyle f:B\to B} 1016:{\displaystyle f:B\to B} 968:{\displaystyle f:B\to B} 711:was holomorphic in each 335:02:14, 12 Feb 2005 (UTC) 183:12:09, 23 Oct 2004 (UTC) 145:project's priority scale 1959:pseudoholomorphic curve 1955:hermitian vector bundle 546:spectrum of an operator 393:18:30, 1 Mar 2005 (UTC) 381:18:30, 1 Mar 2005 (UTC) 369:18:30, 1 Mar 2005 (UTC) 106:WikiProject Mathematics 3407:C-Class vital articles 3267: 3241: 3160:botanical nomenclature 2787: 2671: 2445: 2377:Holomorphic at a point 2213:number of variables. 2103:real analytic function 1932: 1881: 1852: 1830: 1727:actually works in any 1545: 1520: 1429: 1360: 1277: 1163: 1132: 1067: 1017: 969: 937: 896: 849: 815: 765: 732: 705: 651: 591: 529: 477: 2788: 2672: 2446: 1998:analytic continuation 1933: 1882: 1853: 1831: 1546: 1500: 1430: 1361: 1278: 1136: 1105: 1068: 1018: 970: 938: 897: 850: 795: 766: 733: 731:{\displaystyle z_{i}} 706: 652: 571: 530: 478: 353:It means a sequence z 36:level-4 vital article 2742: 2626: 2404: 2309:holomorphic function 2223:holomorphic function 2144:holomorphic function 2098:holomorphic function 1951:holomorphic sections 1895: 1862: 1840: 1804: 1482: 1374: 1293: 1087: 1045: 995: 947: 906: 863: 792: 742: 715: 665: 568: 499: 458: 129:mathematics articles 1994:germ (mathematics)s 1775:Nice work, Linas! - 1438:associative algebra 1272: 1247: 1225: 281:This article says: 2907:Analytic functions 2783: 2667: 2441: 2000:(in the conext of 1928: 1877: 1848: 1826: 1764:Fréchet derivative 1737:Gâteaux derivative 1700:Gâteaux derivative 1696:Fréchet derivative 1671:Gateaux derivative 1541: 1425: 1356: 1273: 1251: 1226: 1204: 1063: 1013: 965: 933: 892: 845: 761: 728: 701: 647: 525: 473: 162:Logarithm question 98:Mathematics portal 45:content assessment 3360:...</math: --> 3219:wherein he says: 3217:Richard Borcherds 2921:comment added by 2886: 2869:comment added by 2305:analytic function 2269:analytic function 2219:analytic function 2140:analytic function 2094:Analytic function 2065:Analytic function 1889:complex structure 1457:analytic function 159: 158: 155: 154: 151: 150: 3449: 3361: 3345: 3338: 3323: 3317: 3313: 3307: 3292: 3185:Overly technical 3170: 3169: 3168: 3136: 3130: 3129: 3128: 3111:Michael Goodyear 3095:Michael Goodyear 3092: 3054: 3037:Michael Goodyear 2977:Michael Goodyear 2943:Michael Goodyear 2933: 2885: 2863: 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1798: 1531: 1521: 1480: 1479: 1461:Oleg Alexandrov 1405: 1392: 1382: 1377: 1372: 1371: 1337: 1318: 1305: 1291: 1290: 1261: 1236: 1214: 1192: 1179: 1169: 1164: 1141: 1110: 1085: 1084: 1043: 1042: 1033:Oleg Alexandrov 993: 992: 945: 944: 904: 903: 866: 861: 860: 826: 816: 790: 789: 745: 740: 739: 718: 713: 712: 681: 668: 663: 662: 634: 615: 602: 592: 566: 565: 550:Oleg Alexandrov 515: 505: 497: 496: 461: 456: 455: 448: 429: 391:Oleg Alexandrov 379:Oleg Alexandrov 367:Oleg Alexandrov 364: 360: 356: 346: 333:Oleg Alexandrov 291:Oleg Alexandrov 279: 277:What is a seam? 226: 164: 128: 125: 122: 119: 118: 96: 89: 43:on Knowledge's 40: 30: 12: 11: 5: 3455: 3453: 3445: 3444: 3439: 3434: 3429: 3424: 3419: 3414: 3409: 3399: 3398: 3395: 3394: 3304:In this case, 3283: 3280: 3205: 3202: 3186: 3183: 3155: 3154: 3153: 3152: 3151: 3150: 3149: 3148: 3147: 3146: 3145: 3144: 3032: 3012: 3011: 3010: 3009: 3008: 3007: 2985: 2938: 2937:Disambiguation 2935: 2908: 2905: 2904: 2903: 2859: 2856: 2855: 2854: 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2114: 2111: 2108: 2104: 2099: 2095: 2091: 2090: 2087: 2084: 2079: 2078: 2077: 2076: 2073: 2064: 2058: 2055: 2050: 2049: 2048: 2047: 2044: 2041: 2036: 2035: 2030: 2026: 2022: 2018: 2017: 2016: 2013: 2008: 2007: 2003: 1999: 1995: 1990: 1989: 1986: 1982: 1978: 1973: 1972: 1971: 1970: 1967: 1962: 1960: 1956: 1952: 1948: 1944: 1939: 1923: 1913: 1903: 1899: 1890: 1872: 1810: 1807: 1795: 1782: 1778: 1774: 1773: 1772: 1769: 1765: 1761: 1760: 1759: 1758: 1757: 1756: 1755: 1754: 1746: 1742: 1738: 1734: 1730: 1726: 1725:Fréchet space 1722: 1721: 1720: 1719: 1718: 1717: 1709: 1705: 1701: 1697: 1693: 1689: 1688: 1687: 1686: 1685: 1684: 1679: 1676: 1672: 1668: 1667:Frechet space 1664: 1660: 1659: 1658: 1657: 1652: 1649: 1645: 1640: 1639: 1638: 1637: 1633: 1629: 1624: 1623: 1619: 1615: 1614: 1613: 1612: 1609: 1605: 1601: 1600: 1597: 1592: 1591: 1588: 1573: 1569: 1564: 1563: 1562: 1559: 1554: 1536: 1532: 1526: 1522: 1511: 1508: 1505: 1501: 1497: 1491: 1485: 1478: 1477: 1475: 1474: 1473: 1472: 1471: 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453: 452:Banach spaces 445: 441: 437: 433: 425: 424: 420: 416: 412: 408: 407: 406: 404: 401: 392: 388: 387: 386: 380: 376: 375: 374: 368: 352: 351: 350: 343: 334: 330: 329: 328: 327: 326: 325: 317: 316: 315: 314: 313: 312: 306: 302: 298: 297: 296: 295: 294: 292: 284: 283: 282: 276: 272: 268: 264: 259: 258: 255: 251: 247: 242: 241: 237: 236: 231: 230: 229: 223: 217: 213: 209: 204: 203: 202: 201: 196: 193: 189: 188: 187: 186: 182: 178: 177: 176: 174: 170: 161: 146: 140: 137: 136: 133: 116: 112: 108: 107: 99: 93: 88: 86: 83: 79: 78: 74: 71: 68: 65: 61: 56: 52: 46: 38: 37: 27: 23: 18: 17: 3367:MOS:STRAIGHT 3364: 3355: 3348: 3342: 3335: 3326: 3303: 3285: 3268: 3262: 3257: 3255: 3249: 3242: 3236: 3232: 3228: 3224: 3222: 3210: 3207: 3188: 3156: 3014: 3013: 2961: 2940: 2917:— Preceding 2914: 2910: 2893:46.193.1.224 2865:— Preceding 2861: 2811:JamesBWatson 2802: 2798: 2794: 2735: 2731: 2727: 2723: 2719: 2679:JamesBWatson 2619: 2615: 2611: 2607: 2603: 2585: 2581: 2578: 2575: 2559: 2524:The 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rated 3351:MOS:MATH 3250:-morphic 3233:-morphic 3158:area of 2919:unsigned 2879:contribs 2867:unsigned 2838:Gwideman 2698:Gwideman 2588:Gwideman 2392:Amorette 2040:SammyBoy 1966:SammyBoy 1665:and the 3139:Quondum 3102:Quondum 3061:Quondum 3022:Quondum 2989:WP:NAMB 2968:Quondum 1694:is the 405:Jordan 41:C-class 3371:Beland 3319:thinsp 3309:hairsp 2610:, but 2171:MarSch 2083:MarSch 2054:MarSch 1289:where 192:MarSch 47:scale. 3369:. -- 3229:holo- 3098:added 2992:here. 2568:JSto 2470:merge 2344:Lunch 2334:Lunch 2012:linas 1953:of a 1938:). 1858:with 1777:Lethe 1768:linas 1741:Lethe 1704:Lethe 1675:linas 1648:linas 1628:Lethe 1608:linas 1596:linas 1587:linas 1568:Lethe 1558:linas 1448:linas 977:linas 773:linas 738:(and 539:with 485:linas 421:Ricky 398:bit. 301:Lethe 169:Lethe 28:This 3375:talk 3341:x + 3339:and 3334:x + 3195:talk 3176:talk 3115:talk 3078:talk 3041:talk 2998:talk 2947:talk 2927:talk 2897:talk 2875:talk 2842:talk 2815:talk 2732:Real 2702:talk 2683:talk 2614:and 2592:talk 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1055:B 1052:: 1049:f 1011:B 1005:B 1002:: 999:f 963:B 957:B 954:: 951:f 931:B 925:B 922:: 919:) 916:z 913:( 910:g 889:C 881:C 877:: 872:k 868:g 843:) 840:z 837:( 832:k 828:g 822:k 818:a 812:N 807:0 804:= 801:k 759:B 751:k 747:a 724:i 720:z 698:C 689:N 684:C 679:: 674:k 670:f 645:) 640:N 636:z 632:, 626:, 621:1 617:z 613:( 608:k 604:f 598:k 594:a 588:N 583:0 580:= 577:k 541:z 521:n 517:z 511:n 507:a 469:n 464:C 434:( 428:n 413:( 363:0 359:n 355:n 265:( 248:( 210:( 147:. 53::

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