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  "body": "\n## Summary\n\nAn exponential function on a [[semigroup|0XMEM406JM]] $S$ is a complex-valued function $m$ with $m(xy) = m(x) m(y)$ for all $x, y \\in S$.\nOn the additive group $\\R$, the continuous ones are the functions $x \\mapsto e^{\\lambda x}$ with $\\lambda \\in \\C$, and the zero function.\nExponentials are the multiplicative solutions of Cauchy's equation on $S$, and they generalize the [[characters of abelian groups|H1WNVN5V7F]].\nMany functional equations on groups and semigroups have solutions built from a few exponentials, such as $\\frac{1}{2}(m(x) + m(x^{-1}))$ for d'Alembert's equation.\n\n## Explanations\n\n### L1 human\n\nAn exponential function turns the operation of the semigroup into multiplication of numbers.\nOn the natural numbers under addition, $m(n) = 2^n$ is an example, since $2^{a + b} = 2^a \\cdot 2^b$.\nOn the positive integers under multiplication, $m(n) = n^3$ is an example.\nThese functions play the role that $e^x$ plays on the real line.\n",
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