Talk:Analytic function

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)kk

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)