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  "body": "\n## Definition\n\n*d'Alembert's functional equation* is the [[functional equation|SRAN4EKWS0]]\n$$f(x + y) + f(x - y) = 2 f(x) f(y) \\quad \\text{for all } x, y \\in \\R,$$\nfor an unknown [[function|0EPAT51DXQ]] $f \\colon \\R \\to \\R$.\nA solution is a function $f$ that satisfies this identity for all real $x$ and $y$.\n\n## Remarks\n\nExample: $f(x) = \\cos(2x)$ is a solution, by the product formula $2 \\cos u \\cos v = \\cos(u + v) + \\cos(u - v)$.\nNon-example: $f(x) = 2$ is not a solution, since the left side is $4$ and the right side is $8$.\nSetting $y = 0$ gives $2 f(x) = 2 f(x) f(0)$, so either $f = 0$ or $f(0) = 1$.\nIf $f(0) = 1$, setting $x = 0$ gives $f(-y) = f(y)$, so every nonzero solution is even.\nThe continuous solutions are $f = 0$, $f(x) = \\cos(ax)$ and $f(x) = \\cosh(ax)$ with $a \\in \\R$.\nThe same equation is studied for complex-valued functions on groups, in the form $f(xy) + f(xy^{-1}) = 2 f(x) f(y)$.\n",
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