Bit of a silly question, someone told me that the reason even functions are called ‘even’ and odd functions are called ‘odd’ is that all (single-variable) monomials with even powers are even functions and all monomials with odd powers are odd functions. In addition, this implies that the MacLaurin series (if it exists) of the odd part of a function is a linear combination of monomials with odd powers and similarly for the even case.
I’ve done a google search about this, but I haven’t had much success. Does anyone know if this the actual origin of the names ‘even’ and ‘odd’?
Probably because polynomials and convergent power series in which all terms have even degree are even functions, and similarly for odd.
EVEN FUNCTION. Functiones pares is found in 1727 in “Problematis traiectoriarum reciprocarum solutio,” presented to the Petersburg Academy in July 1727 by Leonhard Euler:
Primo loco notandae sunt functiones, quas pares appello, quarum haec est proprietas, ut immutatae maneant, etsi loco $x$ ponatur $-x$. [In the first place are noted functions, which I call even, of which there is this property, that they remain unchanged if in place of $x$ is put $-x$.]
end of quote
I know that the term “even” derives from the fact that the Taylor series of an even function contains only even powers, and so for the odd functions.