d'Alembert's functional equation

Summary

d'Alembert's functional equation is the functional equation $f(x + y) + f(x - y) = 2 f(x) f(y)$ for an unknown function $f \colon \R \to \R$. It is the addition rule of the cosine, written as an equation. Its continuous solutions are $f = 0$, $f(x) = \cos(ax)$ and $f(x) = \cosh(ax)$ with $a \in \R$. Versions on groups, with $x - y$ replaced by $x y^{-1}$, are a main topic of the theory of functional equations on groups.