Talk:Even and odd functions

Extensions?
are there extensions of these simple idas to higher dimensions? --achab 16:44, 28 May 2007 (UTC)
 * This concept is in no way bound to real numbers. The definition can be applied verbatim to any function $$f : G \to H$$, where $$(G,+)$$ and $$(H,+)$$ are arbitrary groups.  Thus it also works without any change for vector spaces of any dimension.  --131.188.3.21 (talk) 11:47, 16 June 2009 (UTC)

Origin
Okay, so what *is* the original of the terms even/odd, if not from Taylor series? It's certainly not just "coincidence", as no sane person would keep the term "odd" for the even-powered monomials or vice versa. 216.167.142.196 04:46, 17 November 2005 (UTC)


 * Originally, the word "even" comes from "level", while "odd" comes from "sticking out".  says that the first instance of "even function" was in 1727 by Leonhard Euler, "odd function" in 1819 . — Omegatron 21:56, 17 January 2007 (UTC)

Starting with quotients, there is a word missing in the properties.

Why?
The choice of even and odd seems arbitrary, I've never seen it explained anywhere. Could somebody explain the motivation for defining even and odd functions? --yoshi 05:32, 23 January 2006 (UTC)
 * Even when you divide in half you have mirror image on each side of divisor (so it equals itself). Odd you're one man short which makes people sad.  To signify sadness/negativity, we use the minus sign. 69.143.236.33 06:29, 12 October 2007 (UTC)
 * As far as I'm aware, the terms odd and even are derived from the exponents of some basic odd and even functions ; x2 has the property that f(x)=f(-x) -- i.e. x2=(-x)2. Similarly with x4, x6 and so on. Since these have even exponents, all other functions which have this property are referred to as even. The opposite is true for x, x3, x5 and so on, so they are referred to as odd functions.--86.165.254.170 (talk) 16:08, 6 May 2008 (UTC)

Negative exponents
So is xn an odd function if n is a negative odd integer (even if it's undefined at zero)? — Loadmaster 20:03, 17 January 2007 (UTC)
 * Yes. --Spoon! 03:33, 13 March 2007 (UTC)

Proofs
The proprieties listed here http://en.wikipedia.org/wiki/Even_and_odd_functions#Basic_properties are quite plain.. Somoene should add a short proof for each propriety. —Preceding unsigned comment added by stdazi (talk • contribs)
 * I'm not sure that's a good idea. The properties are so simple, I think the proofs can be left to the reader.  Perhaps a proof or two could be given, but we don't need one for every property. Doctormatt 23:07, 11 August 2007 (UTC)

Definitions
I think we should make the definitions of odd and even functions more strict. My suggestions are: Let $$f:A\to\mathbb{R}$$ where $$A\subseteq\mathbb{R}$$ ƒ is even if and only if $$f(x)=f(-x)$$ for all $$x\in A$$ Similarily ƒ is odd if and only if $$f(x)=-f(-x)$$ for all $$x\in A$$ DanielEriksson87 15:06, 11 September 2007 (UTC)

real-valued
The definition in the article restricts f to be real valued. There is no need for this restriction. Actually it is often usefull to also consider complex valued even or odd functions. --131.188.56.77 (talk) 09:16, 16 June 2009 (UTC)

I think for Complex function you have to use the conjugate.--Royi A (talk) 20:12, 24 September 2009 (UTC)

Geometric properties?
A graph of an even function has reflection symmetry, and a graph of an odd number has rotation symmetry. 99.185.0.100 (talk) 00:41, 10 May 2015 (UTC)

أسامة حسن حسين
أسامة حسن حسين هو طالب في مدرسة (ٍStem).يقال أن السيد أسامة هو اكثر شخص اكتأبا ولكن أنا أنفي ذلك نفي بات. — Preceding unsigned comment added by 197.38.79.23 (talk) 17:34, 8 December 2016 (UTC)

Integral of an odd function
For an odd function $$f(x)$$ the integral $$\int_{-A}^{A} f(x) dx = 0$$ even when $$A = \infty$$. The statement that $$A$$ must be finite is incorrect. My contribution to this paragraph has this citation, the original content appears to have no citation for to support its incorrect assertion.
 * This is not true. $$\int_{-\infty}^{+\infty} x dx$$ does not even exist. If one wants to make sense of that, one must generally resort to Cauchy principal values. This is not contradicted by the link you gave, which deals with bounded intervals only. The subtle point is the meaning of integrability over the whole real line.--Jasper Deng (talk) 07:45, 11 December 2017 (UTC)

Teddy theorem
I have reverted twice the insertion of a section called "Teddy theorem" for the following reasons. In summary, the edits add nothing else than unreliable references and a theorem name that is WP:OR. They cannot be accepted without breaking Wikipedia policies. D.Lazard (talk) 15:25, 7 November 2019 (UTC)
 * One of the two provided sources is a web site that cannot be used as a source per WP:ELNO. The name "Teddy theorem" seems to not be used in this web site.
 * The second source is not a textbook, but a self-biography of a mathematician. The section refers specifically to a chapter called "Preparing the war". I have not access to this chapter, but one may guess that the term "Teddy theorem" refers to the relationship between the author and other soldiers about mathematics. In any case, this does not means that the name is used elsewhere for this result. So, the name is not notable enough for appearing in Wikipedia.
 * Presented as it is, the theorem is wrong as supposing implicitely that all functions are infinitely differentiable. The way of preenting the result in the section "In calculus" is also confusing, but the title of the section lets the reader supposing that the functions considered are differentiable. However, it would be better to mention that in the section.

I think the convention in definition is to put f(-x) to the left of the equal sign.
That's how it is in my textbooks and at Wolfram. 69.5.112.154 (talk) 23:36, 2 May 2023 (UTC)