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  "body": "\n## Definition\n\nLet $S$ be a [[semigroup|0XMEM406JM]], with its operation written as $(x, y) \\mapsto xy$.\nAn *exponential function* on $S$, or *exponential*, is a [[function|0EPAT51DXQ]] $m \\colon S \\to \\C$ such that\n$$m(xy) = m(x)\\, m(y) \\quad \\text{for all } x, y \\in S.$$\n\n## Remarks\n\nExample: on $S = (\\R, +)$, $m(x) = e^{(1 + 2i) x}$ is an exponential, since $e^{(1 + 2i)(x + y)} = e^{(1 + 2i) x} e^{(1 + 2i) y}$.\nNon-example: on $S = (\\R, +)$, $m(x) = 2 e^x$ is not an exponential, since $m(0 + 0) = 2$ while $m(0) m(0) = 4$.\nThe zero function is an exponential; some authors exclude it.\nIf $S$ is a group and $m$ is not identically zero, then $m(e) = 1$, $m$ never vanishes, and $m$ is a homomorphism into the multiplicative group $\\C^{\\times}$.\nAn exponential on an abelian group whose values all have absolute value $1$ is a [[character|H1WNVN5V7F]] of the group.\n",
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