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  "body": "\n## Summary\n\nd'Alembert's functional equation is the [[functional equation|SRAN4EKWS0]] $f(x + y) + f(x - y) = 2 f(x) f(y)$ for an unknown function $f \\colon \\R \\to \\R$.\nIt is the addition rule of the cosine, written as an equation.\nIts continuous solutions are $f = 0$, $f(x) = \\cos(ax)$ and $f(x) = \\cosh(ax)$ with $a \\in \\R$.\nVersions on groups, with $x - y$ replaced by $x y^{-1}$, are a main topic of the theory of functional equations on groups.\n\n## Explanations\n\n### L1 human\n\nThe [[cosine|08WFZP6P89]] satisfies $\\cos(x + y) + \\cos(x - y) = 2 \\cos x \\cos y$.\nThe equation turns this identity around and asks for all functions with the same property.\nFor example, $\\cosh x = (e^x + e^{-x})/2$ also satisfies it.\nUp to stretching the variable and the trivial solution $0$, the cosine and the hyperbolic cosine are the only continuous solutions.\n",
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