---
tag: QWQ45PQTZT
kind: concept
name: Exponential functions on semigroups
msc: [39B52, 20M15, 43A40]
sources:
  - url: https://en.wikipedia.org/wiki/Character_(mathematics)
---

## Summary

An exponential function on a [[semigroup|0XMEM406JM]] $S$ is a complex-valued function $m$ with $m(xy) = m(x) m(y)$ for all $x, y \in S$.
On the additive group $\R$, the continuous ones are the functions $x \mapsto e^{\lambda x}$ with $\lambda \in \C$, and the zero function.
Exponentials are the multiplicative solutions of Cauchy's equation on $S$, and they generalize the [[characters of abelian groups|H1WNVN5V7F]].
Many 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.

## Explanations

### L1 human

An exponential function turns the operation of the semigroup into multiplication of numbers.
On the natural numbers under addition, $m(n) = 2^n$ is an example, since $2^{a + b} = 2^a \cdot 2^b$.
On the positive integers under multiplication, $m(n) = n^3$ is an example.
These functions play the role that $e^x$ plays on the real line.