Talk:Functional equation

Discussion in 2004

I don't really agree with this. I'm not familiar with any functional equations qua equations between linear functionals, for one thing.

Charles Matthews 18:38, 6 Apr 2004 (UTC)

I'm inclined to agree with you. The first item was an example of what I've always thought the term functional equation means, but it's only one example; there are many others. I could list examples, and maybe with effort I could list interesting examples (such as the one that was mentioned before I erased this page), but I'm not sure I know an actual definition of this concept. Michael Hardy 19:00, 6 Apr 2004 (UTC)

There is a little more content here, now. It might be better to merge this all into functional (mathematics), at some point in the future. Maybe just see how it goes. Charles Matthews 08:19, 14 Apr 2004 (UTC)

Actually, there already is functional, with an equally poor definition of a functional equation. Mikkalai 15:16, 14 Apr 2004 (UTC)

I know - functional equation used to be a redirect to functional. You say a poor definition. It is certainly quite a challenge to give a complete and formal definition - which would also include all interesting examples, of course. I don't know - there could be functional equations involving convolution? Charles Matthews 16:09, 14 Apr 2004 (UTC)

What about Gamma, Beta and so on? I am not an expert, but for me, functional equations regarded those functions... ?? Nonsense? :/ Pfortuny 15:11, 15 Apr 2004 (UTC)

OK, the gamma function is defined (more-or-less) by some functional equation(s). There is a difficulty, felt at operator for example, that to be a good encyclopedia article you need some abstract talk, as well as examples.

Tell you what, time to look in some rival products. Take the big Soviet encyclopedia of maths. The editorial notes in the English translation actually quarrel with the original article. Says that typical are

f(x + y) = f(x) + f(y) (Cauchy equation)

F(az) = aF(z)(1 − F(z)) (Poincaré equation)

G(x) = λ^-1G(G(λz)) (chaos theory, scaling)

also gamma function, Riemann zeta, Yang-Baxter and things like

f((x + y)/2) = (f(x) + f(y))/2 (Jensen)

g(x + y) + g(x − y) = 2g(x)g(y) (d'Alembert)

f(h(x)) = cf(x) (Schröder)

f(h(x)) = f(x) + 1 (Abel).

And more. I'll look in the Japanese encyclopedia, too.

Charles Matthews 15:35, 15 Apr 2004 (UTC)

Wow! This is a clear, precise and FAST response. Actually, those are the examples that generally come to mind, aren't they?

Thanks for the job! Pfortuny 15:42, 15 Apr 2004 (UTC)

In that (EDM) under Special Functional Equations (no limiting operations), we get

- discontinuous solution of Cauchy equation by Hamel basis for R over Q (Hamel-Lebesgue)

- Jensen equation, when solutions must be continuous

- general addition theorem, result of Picard in the complex plane I - d'Alembert equation, continuous solutions via cosh and cos

- stuff on Schröder, Abel.

So, looks like there really are half-a-dozen basic examples.

Charles Matthews 15:49, 15 Apr 2004 (UTC)

I added some choice words which should help any wayward users exit this page poste haste (do you people ever consider writing something, oh i dunno, useful to people who aren't mathematicians)--naryathegreat 02:46, Jul 11, 2004 (UTC)

I don't expect to fully understand articles on physics, for example. I don't expect it's possible to write articles on physics so that people who have no understanding on physics to understand its content completely. Dysprosia 08:25, 11 Jul 2004 (UTC)

For as little "content" as you've got, you sure as heck better be able to explain it. Second of all, crap isn't true profanity and "enjoys" wide usage in america.--naryathegreat 13:49, Jul 11, 2004 (UTC)

Crap is not an appropriate word to describe encyclopedia articles. I don't know what you're talking about with "my content". Dysprosia 00:47, 12 Jul 2004 (UTC)

Well, I don't know why you assume this site is for Americans, or by Americans, or to be judged by American standards. As you can see, this article has been discussed by various Wikipedians, with the intention of adding to it. If you looked into it, you'd see that the same people write on other topics. Charles Matthews 18:50, 11 Jul 2004 (UTC)

Well, Americans founded it and the servers are in America, so...but anyway, when i was here last, a subject which i understand well consisted of a cryptic paragraph.:It's better now.--naryathegreat 01:05, Jul 14, 2004 (UTC)

Narya, your comments were disrespectful even if not profane. Admittedly, the article was a stub that didn't say much, but it already said so on its face. Michael Hardy 02:12, 14 Jul 2004 (UTC)

To the author of ths page: Power function should be: f(x+y) = f(x) * f(y) (rather then f(xy) on the left)

Solving Functional Equations

Although this part says nothing but the truth, this solution is not complete. $f(x)=ax^{2}$ for any positive a is also a solution (as can be quickly checked). The way this result can be achieved is by induction. I'll try to fix that now in one of my own pages. Jotomicron | talk 17:02, 21 February 2006 (UTC)

I think the error is by assuming $f(0)$ not to be 0, but am not sure. I changed the example to another. If someone corrects the old example feel free to restore it. googl t 14:06, 26 December 2006 (UTC)

In Solving functional equations

the real soluction is

f[x] == (Sqrt[x]*C[1])/Sqrt[2] or f[x] == -(((-1)^(Log[x]/Log[2])*Sqrt[x]*C[1])/Sqrt[2])

por All x Integer

—Preceding unsigned comment added by 201.58.190.135 (talk • contribs)

No, x and f(x) are assumed to be real, so sqrt(x) is undefined. googl t 17:02, 31 December 2006 (UTC)

It is interesting to note that in The Use of Integral Transforms by I.N.Sneddon (pub.: McGraw-Hill (1972)), the author demonstrates solutions of various functional equations, by means of a range of integral transform methods. Hair Commodore (talk) 20:08, 27 January 2009 (UTC)

______________________________________________________________________________________________________________________________________________________ just a little thing for f(x+y)^2=f(x)^2+f(y)^2 f does not exist not just for x=-y but for any real x,y f(x+y)^2=f((x+y)+(-y))^2+f(y)^2 f(x+y)^2=f(x+y)^2+f(-y)^2+f(y)^2 0=f(-y)^2+f(y)^2 this isn't much more complicated and it's more accurate. — Preceding unsigned comment added by 94.159.193.2 (talk) 21:18, 21 May 2015 (UTC)

Examples

Hi. If I'm not wrong $f(x)={f(x+1) \over x}\,\!$ it is a function not equation. Am I right ?--Adam majewski (talk) 09:30, 2 November 2008 (UTC)

f(x) is a function, f(x) = f(x+1)/x is an equation, indeed a functional equation. RobHar (talk) 16:01, 2 November 2008 (UTC)

Yes. You are right, I was wrong ( probably blind) (:-)) --Adam majewski (talk) 16:31, 2 November 2008 (UTC)

Babbage equation $f^n(x)=x, \forall x\in S$ is one classic example. — Preceding unsigned comment added by Scumat (talk • contribs) 03:24, 31 May 2011 (UTC)

I removed the virtually incomprehensible inserts of 8 September 2008, by Math1353, as decidedly non-standard and unhelpful in illustrating the conventional canon of functional techniques. A committed proponent of the stuff might try to incorporate them, with lots of work to make them pedagogically accessible, into a new stub---or else a new section. But there should be some consensus reached beforehand that this is indeed material that can be usefully flung on the novice who comes to this page to learn what functional equations are all about. Would anyone here hurl this to their students at their first exposure to the subject? I concur with the move of User:124.170.241.132 four months ago to protect the reader, and strongly disagree with unwarranted revert thereof. Cuzkatzimhut (talk) 19:59, 20 May 2014 (UTC)

name of the area of mathematics

What would be the name of the branch of mathematics (https://en.wikipedia.org/wiki/Mathematics_Subject_Classification#First-level_areas) (or say, the typical name of a college class) where functional equations would be studied? Can a little more of that context be added to the intro? DKEdwards (talk) 19:26, 27 February 2023 (UTC)

The first level area 39 is "Difference (equations) and functional equations”. Nevertheless, functional equations are generally considered to belong to mathematical analysis. In the Wikipedia category system (see the bottom of the article), the category Functional equations is a subcategory of the category Mathematical analysis. Nevertheless, as functional equations are used and considered in many branches of mathematics, it could be misleading to add a rigid classification in the introduction. D.Lazard (talk) 21:09, 27 February 2023 (UTC)

Ok, I think I see what you mean. So the idea is if a reader wants to know the higher level context a topic is typically classified under (for example, if they want to find a college course on the subject), they should go to the bottom of the page, click the name beside "Category:", then go to the bottom of that page, and see what is listed after that page's "Category:" line.

It's nice that it kind of works to do that. But I think that most people won't know that process, and classification context can be very helpful for understanding the scope of a subject etc. I think the description you just gave me above is just fine and would be appropriate in the article. It doesn't need to be a single rigid category... just a little more context. I know some WP articles have really nice drop-down tables at the bottom showing classification and related topics. I wish all math articles had those tables, but I don't know how hard they are to incorporate. DKEdwards (talk) 20:37, 4 March 2023 (UTC)