---
tag: Q95N549AE3
kind: definition
name: Periodicity equation
msc: [39B22]
defines: YZNFWJAKA4
uses: [SRAN4EKWS0, 0EPAT51DXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Periodic_function
---

## Definition

Let $p$ be a nonzero real number.
The *periodicity equation* with period $p$ is the [[functional equation|SRAN4EKWS0]]
$$f(x + p) = f(x) \quad \text{for all } x \in \R,$$
for an unknown [[function|0EPAT51DXQ]] $f \colon \R \to Y$, where $Y$ is a set.
The solutions are the [[periodic functions|0ATF6Q0633]] with period $p$.

## Remarks

Example: $f(x) = \cos(\pi x)$ is a solution for $p = 2$, since $\cos(\pi x + 2\pi) = \cos(\pi x)$.
Non-example: $f(x) = x^2$ is not a solution for any $p \ne 0$, since $f(0 + p) = p^2 \ne 0 = f(0)$.
With $q = \abs{p}$, the general solution is $f(x) = h\big(x - q \floor{x/q}\big)$, where $h \colon [0, q) \to Y$ is arbitrary.
Every solution for $p$ is also a solution for each nonzero integer multiple $np$.
The analogous equation $f(x + p) = -f(x)$ is called the antiperiodicity equation; its solutions have period $2p$.