Jacobi Elliptic Functions sn cn dn

Compute the Jacobi elliptic functions sn(u,k), cn(u,k), dn(u,k) from argument u and modulus k. Also shows derived functions sd, cd, nd, the parameter m=k^2, and an sn graph.

Input

Enter argument u and modulus k to compute the Jacobi elliptic functions sn, cn, dn and derived functions using the arithmetic-geometric mean (AGM).

Argument u

Argument of the elliptic functions. Enter a real number.

Modulus k

Modulus k (0 to 1 is assumed). The parameter is m = k squared.

Result

sn(u, k) where u = 1, k = 0.5

0.8226355781

cn(u, k)

0.5685689981

dn(u, k)

0.9114920057

Parameter m = k squared

0.25

Graph of sn(u, k)

List of basic and derived functions

Function	Value
sn(u, k)	0.8226355781
cn(u, k)	0.5685689981
dn(u, k)	0.9114920057
sd = sn / dn	0.9025154066
cd = cn / dn	0.6237783706
nd = 1 / dn	1.0971023265

How it works

The Jacobi elliptic functions sn(u,k), cn(u,k), dn(u,k) are computed from argument u and modulus k using a descending transformation based on the arithmetic-geometric mean (AGM).
Modulus convention: this tool takes the modulus k as input, and the parameter m is m = k^2. Some references use m directly as the modulus, so check which convention you need.
The identities sn^2 + cn^2 = 1 and dn^2 + k^2 sn^2 = 1 hold. Derived functions are defined as sd = sn/dn, cd = cn/dn, and nd = 1/dn.
When k = 0 the functions reduce to sn = sin u, cn = cos u, dn = 1. When k = 1 they reduce to sn = tanh u, cn = dn = sech u.
The range 0 <= k < 1 is assumed. If you enter |k| >= 1, the result is approximated with the degenerate k = 1 form.

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