---
tag: FF9ZPCZ59K
kind: concept
name: Elliptic function
msc: ["33E05", 30D30]
specializes: [0ZZ7ZYDR8W]
sources:
  - url: https://en.wikipedia.org/wiki/Elliptic_function
same_as:
  wikidata: Q938102
  wikipedia: Elliptic function
---

## Summary

An elliptic function is a meromorphic function on the complex plane with two periods that are linearly independent over the real numbers.
It is the same as a meromorphic function on a complex torus $\C / \Lambda$.
Elliptic functions arose from the inversion of elliptic integrals, and they are the analytic side of the theory of elliptic curves.

## Explanations

### L1 human

The sine function repeats when you move by $2\pi$ along the real axis.
An elliptic function repeats along two different directions, so its values are determined by one parallelogram.
Gluing the opposite sides of this parallelogram gives a torus, the surface of a doughnut, and the function lives on that torus.
A nonconstant elliptic function must have poles: without poles it would be bounded, since it is bounded on one parallelogram, and a bounded entire function is constant.
The [[Weierstrass elliptic function|361NAK68QK]] is the basic example, with a double pole at each lattice point.