---
tag: MDWN6E58AW
kind: concept
name: Cauchy's power equation
msc: [39B22]
specializes: [SRAN4EKWS0]
sources:
  - url: https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation
---

## Summary

Cauchy's power equation is the [[functional equation|SRAN4EKWS0]] $f(xy) = f(x) f(y)$ for an unknown function $f \colon (0, \infty) \to \R$.
It asks which functions turn products into products, that is, which functions are multiplicative.
Its continuous solutions are the zero function and the power functions $f(x) = x^c$ with $c \in \R$.
It is the fourth of Cauchy's basic equations, beside the additive, exponential and logarithmic ones.

## Explanations

### L1 human

Scaling the input by a factor scales the output by a matching factor.
The area of a square of side $x$ is $x^2$, and doubling the side multiplies the area by $2^2 = 4$.
The function $f(x) = x^2$ thus satisfies $f(xy) = f(x) f(y)$.
This equation is the rule behind scaling laws in geometry and physics.