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

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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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     author = {Kannappan, PL},
     title = {A {Functional} {Equation} for the {Cosine}},
     journal = {Canadian mathematical bulletin},
     pages = {495--498},
     year = {1968},
     volume = {11},
     number = {3},
     doi = {10.4153/CMB-1968-059-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1968-059-8/}
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