Talk:Holomorphic function

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Logarithm question[edit]

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)
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)

What is a holomorphic function?[edit]

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?

What is a seam?[edit]

This article says:

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

Can some expert clarify what a seam is? Thanks! Oleg Alexandrov 06:10, 11 Feb 2005 (UTC)

Never heard that word before in my life. I'm suspicious. Maybe it's a nonstandard term for branch cut? -Lethe | Talk 09:07, Feb 11, 2005 (UTC)
My motto is: When in doubt, cut it out.
It is a small insignifcant fragment anyway. So what do you think? Oleg Alexandrov 02:14, 12 Feb 2005 (UTC)

"The limit here is taken over all sequences of complex numbers approaching z0," ?[edit]

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)

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...

Yes, this is another way of proceeding about it. Oleg Alexandrov 18:30, 1 Mar 2005 (UTC)

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).

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)

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 bit. 74.104.2.135 03:58, 15 November 2006 (UTC) Jordan

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

Holomorphic functions on Banach spaces[edit]

If I'm not mistaken, one can define the concept of holomorphic functions on Banach spaces, 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 \mathbb{C}^n, I guess), but additional insights, theorems, references, etc. for infinite-dimensional spaces would be welcome. linas 03:49, 20 September 2005 (UTC)

You mean series of the form
\sum a_n z^n
with z 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 spectrum of an operator is nonempty. Oleg Alexandrov 04:22, 20 September 2005 (UTC)
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
\sum_{k=0}^N a_k f_k(z_1,\ldots,z_N)
where each f_k:\mathbb{C}^N\to \mathbb{C} was holomorphic in each z_i (and a_k \in B). 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. linas 00:57, 21 September 2005 (UTC)
The other possibility being
\sum_{k=0}^N a_k g_k(z)
where each g_k:\mathbb{C}\to \mathbb{C} was holomorphic. But that would mean that g(z):B\to B was a one-(complex)-parameter family of maps, and I'm pretty sure that is not what was meant. Rather, the map f:B\to B was itself termed "holomorphic". linas 01:05, 21 September 2005 (UTC)

You wrote:

Rather, the map f:B\to B was itself termed "holomorphic".

For that, you need multiplication on B, don't you? So, you would need a Banach algebra to replace the complex number. Messy business. Oleg Alexandrov 02:00, 21 September 2005 (UTC)

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 f:B\to B is holomorphic if it can be written as a power series in N variables
f(a)=\sum_{k_1=0}^\infty \cdots \sum_{k_N=0}^\infty 
c_{k_1k_2\ldots k_N} a_1^{k_1} a_2^{k_2} \ldots a_N^{k_N}
where a=(a_0,a_1,\ldots,a_N)\in B and B is N-dimensional, and c_{k_1k_2\ldots k_N}\in \mathbb{C} 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 associative algebra, but in order to talk about convergence, one needs a norm, and for that, one needs a Banach algebra. Tadah! I can live with that.
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? linas 04:47, 21 September 2005 (UTC)
I would not be so quick. What you have so far is a fancy analytic function. 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. Oleg Alexandrov 04:54, 21 September 2005 (UTC)
(I slap my forehead). Indeed. I guess I missed the "obvious" definition:
f(a) = \sum_{n=0}^\infty c_n a^n
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. linas 05:26, 21 September 2005 (UTC)
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? 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. -Lethe | Talk 11:24, 21 September 2005 (UTC)

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. linas 13:51, 21 September 2005 (UTC)

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". linas 14:10, 21 September 2005 (UTC)

Ka-ching: [1] has definition. linas 14:13, 21 September 2005 (UTC)

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 Banach space.
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 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. -Lethe | Talk 17:54, 21 September 2005 (UTC)
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 Potts model.) So much math, so little time. linas 03:09, 22 September 2005 (UTC)
Anyway, I'm planning on moving discussions about differentiability from both the Banach space and the Frechet space articles to the article on Gateaux derivative, keeping the multiple similar but not the same definitions in place, where they can be contrasted. linas 13:38, 22 September 2005 (UTC)
I'm all for that. But of course, the derivative that is currently in Banach space is the Fréchet derivative, which is distinct from the Gâteaux derivative, so it needs to be moved there instead. -Lethe | Talk 14:29, 22 September 2005 (UTC)
Also, just for clarity, the derivative in Fréchet space actually works in any TVS (for reasons I'm not perfectly clear on, maybe the TVS has to be locally convex). I made an edit recently to Gâteaux derivative to that effect though, so it'll be clear after you make the move. -Lethe | Talk 14:44, 22 September 2005 (UTC)
I now know more about Banach spaces than I used to. The definition of holomorphic functions on Banach spaces is given in the article Fréchet derivative. Review for clarity and correctness. linas 03:40, 25 September 2005 (UTC)
Nice work, Linas! -Lethe | Talk 05:29, 25 September 2005 (UTC)

(real) derivative is complex linear[edit]

It might be useful to mention that complex differentiability of a map f : \mathbb{C} \rightarrow \mathbb{C} (as currently defined) is also equivalent to saying that if we identify \mathbb{C} with \mathbb{R}^2, the derivative Df(p) is complex linear (i.e. commutes with the complex structure on T_p \mathbb{C} \cong \mathbb{R}^2).

This then makes clear the connection with complex geometry (e.g. holomorphic maps on a Riemann surface, holomorphic sections of a hermitian vector bundle, etc.). One could also make a comment about how this then generalizes nicely to the notion of a pseudoholomorphic curve (as introduced by Gromov in '85, Inv. Math.).

The downside is that this rapidly becomes very technical, and perhaps beyond the scope of an encyclopedia. Any thoughts? SammyBoy 20:20, 18 October 2005 (UTC)

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 have in a standalone section if some people would like to skip it. What do you think? Oleg Alexandrov (talk) 23:54, 18 October 2005 (UTC)
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 presentation. For example, I remember seeing a discussion of germ (mathematics)s and analytic continuation (in the conext of Riemann surfaces) somewhere.
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 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. linas 23:59, 18 October 2005 (UTC)
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. Oleg Alexandrov (talk) 01:21, 19 October 2005 (UTC)
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 page directly? SammyBoy 01:14, 8 November 2005 (UTC)
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. --MarSch 12:39, 8 November 2005 (UTC)

Analytic function[edit]

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 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. -- Taku 10:49, 21 October 2005 (UTC)

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 this is what the article is trying to say. --MarSch 11:20, 21 October 2005 (UTC)
No, I am not suggesting either Analytic function or holomorphic function 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 suggesting is that to merge analytic function here and create a new article real analytic function or something. That reflects the real uses, to my knowledge. -- Taku 13:31, 21 October 2005 (UTC)
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 about them.
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, eventually complex analytic functions and holomorphic functions turn out to be the same thing, but this should not be taken for granted.
As such, merging the articles analytic function and holomorphic function 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 the right conditions are met. Oleg Alexandrov (talk) 14:26, 21 October 2005 (UTC)
So holomorphic is by definition (1 times) complex differentiable? Is holomorphic ever used for other then complex functions? --MarSch 16:47, 21 October 2005 (UTC)
MarSch, at least the current article says that a holomorphic function is complex.
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 holomorphic function. Maybe people are wrong in doing this, but isn't that the reality? For example, [2]. [3]. 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. -- Taku 23:15, 21 October 2005 (UTC)

Of course 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 not 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 number of variables.

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 analytic function. Material specific to complex analysis should go in holomorphic function. I think you never encountered analytic functions outside complex analysis, and that might explain your point of view. Oleg Alexandrov (talk) 02:57, 22 October 2005 (UTC)

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. Oleg Alexandrov (talk) 03:12, 22 October 2005 (UTC)

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. -- Taku 23:27, 22 October 2005 (UTC)

Any concrete suggestions about what to put in analytic function to make it clear that it is not the same as holomorphic? Oleg Alexandrov (talk) 09:35, 23 October 2005 (UTC)

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. -- Taku 23:37, 24 October 2005 (UTC)

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 unique would take place.
As such, you are right that the article analytic function, the way it is now, does not provide too much justification for keeping it separate from holomorphic function. 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. :) Oleg Alexandrov (talk) 03:12, 25 October 2005 (UTC)

extension to infinite dimensions[edit]

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)

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)
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)

Holomorphic at a point[edit]

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)

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)


merge[edit]

one assertion does not an article make. Aaronbrick 22:19, 10 December 2006 (UTC)

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)
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)
It's not exactly a precedent, but I recently wrote Solving quadratic equations with continued fractions.  ;^> DavidCBryant 16:06, 2 January 2007 (UTC)
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)

ln vs log[edit]

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)JSto

I suggest clarifying relationship to Taylor series[edit]

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

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

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.

If one or both of these is correct, perhaps the intro could make this clearer? Gwideman (talk) 02:20, 10 December 2009 (UTC)

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 f which has a formal Taylor series, and for which that Taylor series converges to a function g, but f and g 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 f 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 f(x) = e^{- x^{-2}}, with f(0)=0. It is easy to show that all derivatives at the origin are zero, so the Taylor series is also zero.) JamesBWatson (talk) 14:35, 14 December 2009 (UTC)
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 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? Gwideman (talk) 21:49, 14 December 2009 (UTC)
No, that is not the point. For example the Taylor series about the origin for 1/(1-x) converges to the function only for |x| < 0, although the function is holomorphic on the domain {x ≠ 1}. The point is that for Real 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, but it converges to the wrong function. The example I gave above (f(x) = e^{- x^{-2}}) 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 must converge to the correct function. On the other hand, it is not necessarily the case that it converges throughout the domain of the function, as in the case of 1/(1-x).
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. JamesBWatson (talk) 13:16, 16 December 2009 (UTC)
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. Gwideman (talk) 16:18, 16 December 2009 (UTC)

Holomorphism and Homeomorphism[edit]

What's the difference (similarities) between the two? — Preceding unsigned comment added by Lbertolotti (talkcontribs) 17:38, 7 June 2011 (UTC)

Analytic functions[edit]

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)