Geodesic
A~note on the D'Alembert functional equation
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part V, Tome 47 (1974), pp. 182-183

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It is shown that the group analogue of D'Alembert (cosine) functional equation has solutions which are central and symmetrical functions only in the case of abelian group. Necessary and sufficient conditions for representability of the solution in the form $\frac12(U(x)+U(x^{-1}))$ with a homomorphism $U$ are given. 
@article{ZNSL_1974_47_a17,
     author = {A. L. Rukhin},
     title = {A~note on the {D'Alembert} functional equation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {182--183},
     publisher = {mathdoc},
     volume = {47},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1974_47_a17/}
}
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A. L. Rukhin. A~note on the D'Alembert functional equation. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part V, Tome 47 (1974), pp. 182-183. http://geodesic.mathdoc.fr/item/ZNSL_1974_47_a17/