# When was it found that a function could be decomposed as a sum of even and odd functions?

I'm not sure if it's a good question but I was reading about the decomposition of any function, f(x), as a sum of even and odd functions; f(x) is not an even or odd function.

Is it possible to know when this fact (or, theorem?) was discovered and who found it?

• I imagine most any of the "greats" (Darboux, Baire, Lebesgue, Borel, Hilbert, etc.) who was faced with situations in which this was useful (e.g. maybe early Fourier series work in which decompositions into an even part and and odd part was used) were able to quickly recognize this (if not express it the way we would today), but how far back one can find a reasonably explicit record of such a decomposition does seem somewhat interesting. I can imagine this being "widely known" over a number of years, and maybe occasionally appearing in 1800s literature such as (continued) – Dave L Renfro Aug 20 at 10:37
• The Educational Times (had a widely participated in section on math problems and such), Nouvelles Annales de Mathématiques, Zeitschrift für Mathematik und Physik, Cambridge Mathematical Journal, L'Intermédiaire des Mathématiciens. – Dave L Renfro Aug 20 at 10:54
• Thank you, @DaveLRenfro. You referenced Fourier series which appeared for the first time in 1807. Wouldn't Fourier be familiar with such a decomposition in one way or another? Most of the 'greats' you mentioned were born after 1850. So the question still remains: How far back one can find a reasonably explicit record of such a decomposition? – PG1995 Aug 20 at 21:15

It depends partly on what "function" means. According to a definition prevalent before Dirichlet it meant something expressed by a formula and expandable into a power series. The decomposition claim then is obvious, as long as one introduced the notions of even and odd and related them to even and odd powers. Euler does both in Traiectoriarum Reciprocarum Solutio (1727), XIX-XXI (he uses Latin words pares and impares). Luckily, this is one of the papers translated by Ian Bruce:

"§17. In the first place functions are to be noted that I call even, of which this is the property: that they remain unchanged if $$- x$$ is put in place of $$x$$. All the powers of $$x$$ with even exponents are of this kind, and of fractions with even numerators and odd denominators. Then, any such functions of powers of this kind composed either by addition or subtraction, or by multiplication or division, or hence any such construction raised to sum are likewise even powers, such as $$x^{\frac45}$$, $$(ax^2+bx^{\frac23})^n$$.

§18. In the second place, I take heed of odd functions, which in short produce the negative of these, if $$x$$ is changed to $$-x$$. All the powers of x with odd exponents, such as $$x$$ itself, $$x^3$$, $$x^5$$, etc. are of this kind; or fractions of which the numerators and denominators are odd; also functions composed of these powers either added or subtracted, and which are also dignified to be raised to odd exponents, such as $$x^{\frac45}$$, $$(ax^3+bx^{\frac57})^3$$."

Euler did not have much use for them, or decomposition into them, although, as Barnett notes:

"In fact, the expressions $$(e^x + e^{-x})/2$$ and $$(e^x - e^{-x})/2$$ do make an appearance in Volume I of Euler's Introductio in analysin infinitorum (1745, 1748). Euler's interest in these expressions seems natural in view of the equations $$\cos x=(e^{\sqrt{-1}x} + e^{-\sqrt{-1}x})/2$$ and $$\sqrt{-1}\sin x=(e^{\sqrt{-1}x} + e^{-\sqrt{-1}x})/2$$ that he derived in this text. However, Euler's interest in what we call hyperbolic functions appears to have been limited to their role in deriving infinite product representations for the sine and cosine functions. Euler did not use the word hyperbolic in reference to the expressions... nor did he provide any special notation or name for them.

In a Hacker News discussion Decomposing a function into its even and odd parts David Goldblatt suggested that the idea might go back to earlier times:

"I don't have any references for you right now, but I would be surprised if the usage of the terms doesn't pre-date your citation of Euler by at least several decades. The basic technique for deriving Taylor series is known to date to at least 1671, to the correspondence between James Gregory and John Collins; it also shows up in letters from de Moivre to Johann Bernoulli in 1694 and from Leibniz to Bernoulli in 1708. Newton had a geometric means of deriving the coefficients of power series when he was writing the Principa (published in 1687), as demonstrated by the proof of his tenth proposition, and he included a description of the algebraic technique in an early draft of his Quadrature of Curves (but removed it before publication in 1706)... Perhaps nobody bothered to give the concept a name until Euler in 1727, and perhaps nobody even considered the concept with regard to a more general class of functions until even later".

• Thank you very much! So, it wouldn't be wrong to conclude that the idea of decomposition was around during the time of Euler (1707-1783), and possibly it could pre-date it. It'd also suggest that Fourier should be aware of it before he came up with his Fourier series in 1807. – PG1995 Aug 21 at 2:14
• @PG1995 Miller's Earliest Known Uses of Some of the Words of Mathematics mentions that "even function is found in English in 1809 in a translation of Legendre’s 1792 paper “Mémoire Sur Les Transcendantes Elliptiques”", and Lagrange's Théorie des fonctions analytiques (1797) promoted the power series notion of a function. So it is likely that Fourier was aware of it, although one would have to search his works to be sure, these things do not always happen logically. – Conifold Aug 21 at 3:00
• Thanks a lot! I really appreciate your help. – PG1995 Aug 21 at 3:12