Talk:Analytic function

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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 April 2005 (UTC)[reply]

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 April 2005 (UTC)[reply]

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

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"

C^{\omega }

for analytic functions, vs

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)[reply]

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 $z_{0}$ if its derivative f ' (z) exists at $z_{0}$ and at every point in some neighborhood of $z_{0}$ .

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)[reply]

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)[reply]

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)[reply]

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 $f(x)=Log(x)$ on the set of real numbers R. Now pick $x_{0}$ =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)[reply]

Yes, thank you. I'm quite surprised there are so many good mathematicians here. Siyavash2 (talk) 01:23, 25 December 2008 (UTC)[reply]

Banach Space

I don't see the relevance of the following:

"The set $\scriptstyle A_{\infty }(\Omega )$ 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)[reply]

Ops, you're right, my fault. There was written "...of all bounded analytic functions"; then I added a remark on the space

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)[reply]

Dead link

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

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"

\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)[reply]

You are right, I have updated the article. Because not all $f$ in $C^{\infty }$ are analytic. — Preceding unsigned comment added by 81.20.68.181 (talk) 12:44, 15 February 2017 (UTC)[reply]

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)[reply]

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. --pm a 13:09, 9 February 2015 (UTC)[reply]

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)[reply]

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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)[reply]

You are right, I have updated the article. Because not all $f$ in $C^{\infty }$ are analytic. — Preceding unsigned comment added by 81.20.68.181 (talk) 13:44, 15 February 2017 (UTC)[reply]

about Analytic functions of several variables

I would like to change the section name to Analytic functions of several complex variables. However, this page should also explain real analytic functions, so what we need to do may be an extension of the section. see also Function of several real variables--SilverMatsu (talk) 16:04, 1 February 2022 (UTC)[reply]