Usage Notes
Elliptic functions are doubly periodic, single‑valued complex functions of a single variable that are analytic, except at a finite number of poles. Because of the periodicity, we need consider only the fundamental period parallelogram. The irreducible number of poles, counting multiplicities, is the order of the elliptic function. The simplest, non‑trivial, elliptic functions are of order two.
The Weierstrass elliptic functions, ℘(z, ω, ωʹ) have a double pole at z = 0 and so are of order two. Here, 2 ω and 2 ωʹ are the periods.
The Jacobi elliptic functions each have two simple poles and so are also of order two. The period of the functions is as follows:
Function | Periods |
|---|
sn(x, m) | 4K(m) 2iKʹ(m) |
cn(x, m) | 4K(m) 4iKʹ(m) |
dn(x, m) | 2K(m) 4iKʹ(m) |
The function
K(
m) is the complete elliptic integral, see
ELK, and
Kʹ(
m) =
K(1 –
m).
