# Talk:Analytic function

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Why is analytic function redirected to power series? Phys

Why is analytic map redirected to analytic function? -- Taral 18:50, 30 Aug 2004 (UTC)

Because in mathematics, as the terms are usually used, "map" is a synonym of "function", but the latter term is more frequently used. Michael Hardy 20:52, 30 Aug 2004 (UTC)

## Real versus complex analytic

The article seems to say that one of the differences between real and complex analyticity is that there are real differentiable functions that are not real analytic, while all complex differentiable functions are complex analytic. But I'd say that this is a difference between real and complex differentiability, not real and complex analyticity. For the rest, well done with the rewrite. -- Jitse Niesen 12:02, 13 Feb 2005 (UTC)

You are right. The message I wanted to carry across was that the two beasts (real and complex analytic) are different. I did think what a good thing to write would be. The best thing is of course the example of a real analytic function which is not complex analytic, like f(x, y)=x. However, that would have necessitated talking about analytic functions in many variables before I felt the reader was ready.
Any suggestions on how to improve on this? Oleg Alexandrov 15:46, 13 Feb 2005 (UTC)

## real vs. complex, redux

Hi Linas. Thank you for your recent insertions in that article. I have several remarks though.

(a) The text you inserted in the "definitions" section belongs to holomorphic function, not here. In this article, we talk about analytic functions in general (real, and complex), and going into so much fine detail about complex conjugates brings us offtopic. That is of course very appropriate in the article about holomorphic functions. There, if you note, there is even a paragraph about anti-holomorphic, which has to do with function of the conjugate.

(b) It is not correct that

In particular, one does not get a real analytic function by taking the real part of a complex analytic function.

Actually, one does get a real analytic function (in x and y) by taking the real part of a holomorphic function. Example: x=real(z) is analytic in x and y.

(c) In some places, you forget to say "real analytic", "complex analytic", saying only "analytic". that is confusing, because the very purpose of this article is to say which is what.

I think all of these issues stem again from the fact that you chose to contribute not to the correct article. Would you consider writing the parts which have only to do with complex analyticity in the holomorphic function article? Thanks. Oleg Alexandrov 18:51, 9 Apr 2005 (UTC)

Hi Oleg, I'll fix the error you bring up. Also, I'd prefer to leave the other changes here, rather than putting them in holomorphic. I suppose I should explain why: For the first part, I wanted to capture that the series expansion is in one variable z and not two variables z and z-star. I tried to use a minimum of words to say this while still being clear; my apologies if I failed. Making a similar statement in the article on holomorphic would get lost (in part because holomorphic is a different concept which "accidentally" means the same thing as analytic for complex functions). So I picked this article, and not that, after careful consideration. :)
Also, the bit about harmonic ... again, this seemed to be the better article to bring this up, as this article really does try to distinguish real and complex harmonic functions; a lack of distinction often leads to confusion, e.g. in Riemann surfaces in particular. Here, its trivial to see that a complex analytic function is harmonic, because the series expansion clearly doesn't depend on z-star. This idea gets lost, gets opaque ad unclear, if one sticks to holomorphic functions. So putting it in the section on disambiguating real and complex analytic functions seemed the right way to go. linas 19:07, 9 Apr 2005 (UTC)
The point being, I guess, that there is a tremendous service to the readership in clearly distinguishing real and complex analytic functions, and taking some pains in pointing out the common pitfalls and fallacies. Not only have I seen others fall into this trap, I know I have as well ... its all too easy to make trivial assumptions: "Oh yeah, I know this, analytic, this is a simple concept ... real part .. yeah that's trivial ..." and oops, one is thus lead to fallacies which can be hard to get out of. linas 19:25, 9 Apr 2005 (UTC)
The part about the conjugate in the series expansion, I need to think more about. The part about harmonic and stuff, maybe you could consider putting it into a separate section.
And a brief plea. You see, I am a bit picky, because I rewrote "analytic function" from scratch (it was in a really sorry state before). So, I would like to ask you to read very carefully both analytic function and holomorphic function and give it a very careful thought about what should be in both of them.
My vision for analytic function was an article which would explain the concept of analyticity, say for a newbie. That's why your recent additions put me off, they go on tangents onto very delicate details about conjugates and harmonicity, which I feel do not belong in this article.
So, again. Could you think very, very, carefully about what a good article on analytic functions should include, try to make yourself a big picture of this and of holomorphic function. Then let me know what you think. I am sure we can arrive at a satisfactory solution for both of us. Oleg Alexandrov 19:21, 9 Apr 2005 (UTC)
And you missed the fact that this article does not talk about analytic functions in more than one variable. So, what you inserted about "real part of complex analytic function is analytic in x and y" and "complex analytic function is harmonic, but real analytic function is not", is not applicable.
This can be fixed by first talking about real analytic functions in more than one variable, but let us not do that, as then you need to mention complex analytic functions in more than one variable, and power series in more than one variable, and things get complicated. Again, I gave it a careful thought what to include in this article and what to skip, and how to arrange it. And I feel that the article does not have that coherence anymore. Oleg Alexandrov 20:28, 9 Apr 2005 (UTC)
OK,I just got tangled up in other matters. I guess the bit about harmonic functions can be moved to the article on holomorphic functions. Give me a day, or try the move yourself; I just looked at my watchlist as you suggested, and have now gotten distracted elsewhere. I sympathize; as I look at the elementary articles more often, I am starting to notice how poor condition many of them are in.
Anyway, do give readers some intelligence: just the act of talking about complex functions implies real functions in two variables. Again, I think the great service here is to remind the reader that there are pitfalls by mentally glossing over the differences between real and complex analytic functions. I am far more concerned about high-lighting these pitfalls. By contrast, delving deeply into multiple variables is far less of a concern (to me,at this time). linas 20:54, 9 Apr 2005 (UTC)
You missed my point though. In this article analytic functions of two variables are not even defined. Then, it does not make sense to talk about things which were not defined, no matter how smart the reader is. And no, nobody glosses over the difference between real and complex analytic functions, there was a whole section devoted to the differences, even before you added new stuff. Oleg Alexandrov 22:03, 9 Apr 2005 (UTC)

OK, ... I'll see what I can move around.

## Continuing the analysis

Oleg, any chance you'd want to add a see-also section with analytic continuation and germ (mathematics)? I want to rewrite/expand Riemann surfaces, and the current article there has three sections, on analytic continuation, germs, and examples of analytic continuation, that should be moved elsewhere. (e.g. either to this article, or to the article on analytic continuation). Since you are working this topic, would you care to make this move? linas 00:21, 12 Apr 2005 (UTC)

Do you mean, you want to put links to analytic continuation and germ (mathematics) in analytic function? That should certainly be no problem. About the stuff in Riemann surfaces you want to move, I think it would belong to analytic continuation and germ (mathematics) rather than to analytic function.
I do not work on this topic, actually I have quite little time for the moment. I wrote analytic function a while ago because it was in a sorry state.
If you want to really move things from Riemann surfaces, you might consider first posting your intention on its talk page. Maybe the person who put that stuff in there, had some thing in mind when doing so. Oleg Alexandrov 00:39, 12 Apr 2005 (UTC)
So what happened to all this content? The current article still doesn't mention any of the topics above: e.g. more than one variable, and the common pitfalls involved there (except for the single solitary example ${\displaystyle f(z)=z^{*}}$, but there are also more nice, more subtle examples of this kind.) Also, not a breath of analytic continuation or germs or jets shows up here -- these surely deserve at least a single-sentence mention. Also, an explanation for the "abuse of notation" ${\displaystyle C^{\omega }}$ for analytic functions, vs ${\displaystyle C^{\infty }}$. and then fancy sophisticated stuff, like analysis in inf. dim. spaces, and frechet spaces and etc. The current content seems to be like its copied from a freshman college textbook, it fails to actually mention common topics that involve analytic functions...!? 67.198.37.16 (talk) 17:29, 16 April 2016 (UTC)

## The term "non-zero"

I'm not sure this is ambiguous to anyone else, but when you say a function is "non-zero" (in the second point of the "Properties of Analytic Functions" section) to me that would mean that the function is not identically zero (zero for all points in its domain). I know this is not what is intended. Instead, might one say that such a function is "nowhere-zero," "never-zero," or something else more explicit? To me, at least, this clears up what would have been confusion, had I not already known better. --149.43.x.x

Good point. I now put there "nowhere-zero" instead of just "non-zero". Oleg Alexandrov 16:17, 17 July 2005 (UTC)

## Analytic vs holomorphic

See discussion at talk:Holomorphic function where there is a proposal to merge analytic function with holomorphic function. Oleg Alexandrov (talk) 03:04, 22 October 2005 (UTC)

## Intro

This article would be more useful if the intro did more to 'introduce' the reader to the subject. A simple example of an analytic and a non-analytic function? A little discussion of situations when you care whether a function is analytic or not. ike9898 13:42, 27 November 2005 (UTC)

Hi. I gave some examples. Somewhere towards the bottom, there is also link to an example of a smooth function which is not analytic (this is an interesting, but complicated example). Now, when you care if a function is analytic or not is a question to which I don't have a good answer for the moment. Maybe some other people will have ideas.
Thank you for your feedback. I had no idea there were no examples in here (besides the polynomials). Any other suggestions? Oleg Alexandrov (talk) 21:04, 27 November 2005 (UTC)
I suggest the example of the complex conjugate function ${\displaystyle f(z)={\bar {z}}}$. I found this example extremely instructive when I was studying complex analysis. From real analysis, one has the idea that any smooth, well-behaved function with no "kinks" will be differentiable. The conjugate function is as well-behaved as one could hope, but is not analytic, so it makes a stark contrast between the real and complex notions of differentiability.
The conjugate function also provides a good way to introduce the idea of the uniquene analytic extension of a partial function. Since there is only one analytic extension of the identity function from the reals to the complex numbers, and it must be the identity function, it cannot be ${\displaystyle {\bar {z}}}$, which therefore cannot be analytic. -- Dominus 03:48, 23 May 2006 (UTC)
I added that in. The concept of analytic continuation could be an overkill for this article I would think, it may be more appropriate at holomorphic function in my view. Oleg Alexandrov (talk) 04:22, 23 May 2006 (UTC)

## Set of all analytic functions?

Is there a commonly used notation for the set of all real-analytic functions? I think I've seen P or ${\displaystyle {\mathcal {P}}}$ used for this, with "P" implying "polynomial (of inifinite order)", but wonder if this is the accepted convention. linas 03:42, 15 October 2006 (UTC)

## Definition of analytic functions

The definition for analytic functions given in this article conflicts with that of Churchill:

   R. V. Churchill, Complex Variables and Applications,
New York: McGraw-Hill, 1960.


His definition (p. 40) is more simply that a function f of the complex variable z is analytic at a point ${\displaystyle z_{0}}$ if its derivative f ' (z) exists at ${\displaystyle z_{0}}$ and at every point in some neighborhood of ${\displaystyle z_{0}}$.

I am not convinced by the counterexample, because that function, exp(-1/x), is simply "infinitely smooth" at x=0, producing a Taylor series that evaluates to zero everywhere when expanded about x=0. I think this function should be considered analytic at x=0 since derivatives of all orders exist and are zero at that point.

63.195.51.145 00:07, 16 January 2007 (UTC)

There is no conflict. The definition in the book you mention is valid only for complex analytic functions, while the one given here is valid for both real and complex analytic functions. It is true that a complex differentiable function is the same as an analytic function. That is mentioned in this article. Oleg Alexandrov (talk) 04:22, 16 January 2007 (UTC)

I am unsure if I studied this yet in multivariable Calculus or part of Analysis earlier, but it is interesting. I am just curious whether this is the only possible definition of 'analytic function' if you use an abstract logical definition--maybe the mapping of elements of other relevant sets or categories can be analyzed.--Dchmelik (talk) 12:59, 27 February 2009 (UTC)

## Analytic vs everywhere analytic vs entire

The article defines "Analytic on D", and says that "the absolute value function is not analytic". Isn't it analytic on any interval not containing 0? Perhaps "analytic" should be defined to mean "everywhere analytic".

Similarly, if a complex function is everywhere analytic, does that mean it is entire? There is no link from this page to entire, or vice versa.

Finally, for a function to be analytic on D, must the power series about every point ${\displaystyle x_{0}\in D}$ be convergent on D, or just on some neighbourhood of ${\displaystyle x_{0}}$? Could you give an example of a function analytic on D which has a power series which is not convergent on all of D?

Thanks. LachlanA 01:40, 28 January 2007 (UTC)

Thanks for the note. I clarified in the article that the absolute value is not analytic if defined for all numbers.
Yes, a complex analytic function defined everywhere is called entire. I think this detail may be more appropriate at holomorphic function rather than here.
The function
${\displaystyle f(x)={\frac {1}{x^{2}+1}}}$
(mentioned in the article) defined for real x is analytic on R, but its power series expansion is not convergent on all of R (if it were, it would be convergent on all of C, but it is not since it is not defined for all complex numbers). Oleg Alexandrov (talk) 03:40, 28 January 2007 (UTC)

## Logarithm

Sorry for the bad editting here on this discussion page. I'm not really sure how to create a new category on it since I just created a user name. But I was reading this article on Analytic Functions and I noticed that in the Examples section, the logarithm function was given as an example of an analytic function. So let's consider ${\displaystyle f(x)=Log(x)}$ on the set of real numbers R. Now pick ${\displaystyle x_{0}}$=0. Then we can't find a power series representation of Log(x). Can we? So is the logarithm function not analytic then?

Thanks. Siyavash2 01:04, 24 December 2008

No problem on the editing. There should be a "new section" tag at the top of the page. Otherwise placing the section title between a pair of equal signs (two on each side) will also do it. I think the problem is with the domain you have in mind. The article says that the logarithm is "analytic on any open set of [its] domain." But zero is not in the domain of the logarithm. Does this answer your question? Thenub314 (talk) 21:48, 24 December 2008 (UTC)
Yes, thank you. I'm quite surprised there are so many good mathematicians here. Siyavash2 (talk) 01:23, 25 December 2008 (UTC)

## Banach Space

I don't see the relevance of the following:

"The set ${\displaystyle \scriptstyle A_{\infty }(\Omega )}$ of all bounded functions with the supremum norm is a Banach space."

Does this somehow relate to analyticity? Dratman (talk) 19:11, 9 November 2009 (UTC)

Ops, you're right, my fault. There was written "...of all bounded analytic functions"; then I added a remark on the space ${\displaystyle A(\Omega )}$ (of all analytic functions, which is somehow more relevant btw), and lost the word analytic in the operation. Fixed now. --pma (talk) 23:50, 9 November 2009 (UTC)

When ever I click the second reference, I remain on the Wikipedia page. Blackbombchu (talk) 16:31, 22 March 2014 (UTC)

## Alternative Characterizations

"For every compact set K ⊂ D there exists a constant C such that for every x ∈ K and every non-negative integer k the following bound holds"

${\displaystyle \left|{\frac {d^{k}f}{dx^{k}}}(x)\right|\leq C^{k+1}k!}$

Isn't e^{-1/x} an obvious counterexample? It satisfies the bound but is not analytic. Pwrong (talk) 11:49, 31 October 2014 (UTC)

## Content Improvement

The article states:

″ There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. ″


This sentence seems to have no substance. By definition, shouldn't two subsets of analytic functions be similar in some ways and different in others? Unless someone has a strong reason not to, I will delete this section and replace it with something more substantive, such as the difference between the two. Mgibby5 (talk) 23:29, 6 February 2015 (UTC)

Actually this sentence may sound a bit vague. On the other hand, in the introduction we just want to summarize shortly the contents that are developed in the article. Maybe adding some reference to some sections in the article will do. --pma 13:09, 9 February 2015 (UTC)
I welcome input onto what this sentence should be changed to, and references to other sections would be a good start. I think it would be legitimate and interesting to state that real analytic functions are not a subset of complex analytic functions. Thoughts? Mgibby5 (talk) 03:00, 12 February 2015 (UTC)

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## Section: "Alternative characterizations"

In the section it is stated the following:

"If ƒ is an infinitely differentiable function defined on an open set D ⊂ R, then the following conditions are equivalent.

1) ƒ is real analytic."

But infinitely differentiable function is not in general analytic, which is fact emphasized throughout the article. or did I miss the point ?

GeOinWiKi (talk) 17:30, 28 December 2016 (UTC)