---
tag: P071B692JF
kind: concept
name: Jacobi elliptic functions
msc: ["33E05"]
specializes: [FF9ZPCZ59K]
sources:
  - url: https://en.wikipedia.org/wiki/Jacobi_elliptic_functions
same_as:
  wikidata: Q1473526
  wikipedia: Jacobi elliptic functions
---

## Summary

The Jacobi elliptic functions $\operatorname{sn}$, $\operatorname{cn}$ and $\operatorname{dn}$ are [[elliptic functions|FF9ZPCZ59K]] that depend on a modulus $k$ and generalise the sine and cosine.
They come from inverting the incomplete [[elliptic integral|NXMCYGCB59]] of the first kind, through the [[Jacobi amplitude|WDHHJHVP7N]].
They satisfy $\operatorname{sn}^2 + \operatorname{cn}^2 = 1$ and simple differential equations, and they solve the pendulum equation exactly.
Nine further functions such as $\operatorname{sd} = \operatorname{sn}/\operatorname{dn}$ are formed as quotients.

## Explanations

### L1 human

The sine is the inverse of the integral $\int_0^x dt/\sqrt{1 - t^2}$.
If one puts a second factor under the root and inverts $\int_0^x dt/\sqrt{(1 - t^2)(1 - k^2t^2)}$, one gets $\operatorname{sn}$.
For $k = 0$ the functions reduce to $\sin$, $\cos$ and the constant $1$; for $k = 1$ they become $\tanh$, $\operatorname{sech}$ and $\operatorname{sech}$.
In between they look like sine waves that have been flattened at the tops.
Unlike the sine, they also repeat in an imaginary direction, so they are doubly periodic in the complex plane.