Geodesic
A Functional Equation for the Cosine
Canadian mathematical bulletin, Tome 11 (1968) no. 3, pp. 495-498

Voir la notice de l'article provenant de la source Cambridge University Press

It is known [3], [5] that, the complex-valued solutions of (B) (apart from the trivial solution f(x)≡0) are of the form (C) (D) In case f is a measurable solution of (B), then f is continuous [2], [3] and the corresponding φ in (C) is also continuous and φ is of the form [1], (E) In this paper, the functional equation (P) where f is a complex-valued, measurable function of the real variable and A≠0 is a real constant, is considered. It is shown that f is continuous and that (apart from the trivial solutions f ≡ 0, 1), the only functions which satisfy (P) are the cosine functions cos ax and - cos bx, where a and b belong to a denumerable set of real numbers. 
Kannappan, PL. A Functional Equation for the Cosine. Canadian mathematical bulletin, Tome 11 (1968) no. 3, pp. 495-498. doi: 10.4153/CMB-1968-059-8
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[1] 1. Aczel, J., Lectures on functional equations and their applications. (Academic Press, 1966). Google Scholar

[2] 2. Kaczmarz, S., Sur I'equation fonctionelle f(x) + f(x+y) = ϕ(y)f(x + y/). Fund. Math., 6 (1924) 122-129. Google Scholar

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[4] 4. Van Vleck, B., A functional equation for the sine. Ann. Math. 7 (1910)161-165. Google Scholar

[5] 5. Wilson, W.H., On certain related functional equations. Bull. Amer. Math. Soc, 26(1919–1920)300-312. Google Scholar

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