---
tag: 68SHB7Z721
kind: concept
name: Abel equation
msc: [39B12, 26A18, 30D05]
specializes: [BCGBQ3HK26]
sources:
  - url: https://en.wikipedia.org/wiki/Abel_equation
same_as:
  wikidata: Q4666644
  wikipedia: Abel equation
---

## Summary

The Abel equation is the [[iterative functional equation|BCGBQ3HK26]] $\alpha(f(x)) = \alpha(x) + 1$, where $f$ is a given map and $\alpha$ is unknown.
A solution conjugates the map $f$ to the translation $t \mapsto t + 1$.
An injective solution gives a natural way to define non-integer iterates of $f$, by $f^t(x) = \alpha^{-1}(\alpha(x) + t)$.
It is closely related to [[Schröder's equation|VTJV7JVJEA]]: if $\psi$ solves $\psi(f(x)) = s \psi(x)$ with $s > 0$, $s \ne 1$ and $\psi > 0$, then $\alpha = \log_s \psi$ solves the Abel equation.

## Explanations

### L1 human

The Abel equation asks for a counter that goes up by exactly one each time the map $f$ is applied.
For the doubling map $f(x) = 2x$ on the positive numbers, $\alpha(x) = \log_2 x$ is such a counter.
In the new coordinate $\alpha$, applying $f$ becomes simply adding one.
Then half an application of $f$ is adding one half, which in this example gives $f^{1/2}(x) = \sqrt{2}\, x$.