---
tag: CEGF8W64VQ
kind: concept
name: Cauchy's exponential equation
msc: [39B22]
specializes: [SRAN4EKWS0]
sources:
  - url: https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation
---

## Summary

Cauchy's exponential equation is the [[functional equation|SRAN4EKWS0]] $f(x + y) = f(x) f(y)$ for an unknown function $f \colon \R \to \R$.
It asks which functions turn sums into products.
Its continuous solutions are the zero function and the [[exponential functions|0J0FCWV32D]] $x \mapsto a^x$ with $a > 0$.
It is one of the four basic equations of Cauchy, together with the additive, logarithmic and power equations.

## Explanations

### L1 human

Growth by a fixed factor in equal steps is the picture behind this equation.
If a quantity is multiplied by $f(x)$ over a time $x$ and by $f(y)$ over a further time $y$, it is multiplied by $f(x) f(y)$ over the time $x + y$.
For example, $f(x) = 2^x$ satisfies $2^{x + y} = 2^x \cdot 2^y$.
The equation says that this rule alone almost determines the function.