The **Jacobi Elliptic functions** are a way to express the amplitude φ in terms of an elliptic integral *u* and modulus *k*. They share many properties with trigonometric functions and can be thought of as trig function generalizations; In limiting cases where the parameter tends to zero, the Jacobi elliptic functions sn and cn reduce to their trigonometric counterparts: the sine function and cosine function.

They are named after 19th century mathematician Carl Gustav Jacob Jacobi, who contributed to the theory of elliptic functions.

## Applications of Jacobi Elliptic Functions

These functions are mostly noted for their historical importance. Practical applications for Jacobi elliptic functions include:

- Descriptions of pendulum motion,
- Design of electronic elliptic filters,
- Solutions for nonlinear ordinary differential equations.

They also appear in various problems of classical dynamics, electrostatics, and hydrodynamics.

## Types

There are twelve types of Jacobi elliptic functions denoted by pw(u,k), where:

- u is the real-numbered or complex-numbered argument,
- k is the real or complex modulus (or modular angle, or parameter). Note that convention calls for
*m*if you’re using the parameter or*k*if you’re using the modulus (which is the square root of m) or modular angle (defined as m = sin^{2}α), - p and q can be any of the letters c, s, n or d.

The main types are:

- sn(u, k) =
*Jacobi elliptic sine function of modulus k*, defined as sin φ = sin am(u, k). Analogous to the trigonometric sine function. - cn(u, k) =
*Jacobi elliptic cosine function,*defined as cos φ = cos am(u, k). Analogous to the trigonometric cosine function. - tn(u, k) =
*Jacobi elliptic tangent function*: sin φ/cos φ = sn(u, k) / cn (u, k) - dn (u, k) =
*difference function*(the derivative of φ) =

The functions dc, nc, and dc are extensions to imaginary arguments.

## Properties of Jacobi Elliptic Functions

Many of these properties follow from properties of trigonometric functions (Lutovac et al., 2001):

- sn
^{2}(u, k) + cn^{(u, k) = 1} - k
^{2}sn^{2}(u, k) + dn^{2}(u, k) = 1 - sn(u, k) = sn(-u, k)
- cn(u, k) = cn(-u, k)
- sn(0, k) = 0
- sn(K, k) = 1
- cn(0, k) = 1
- cn(K, k) = 0

## References

Lutovac, M. et al., (2001). Filter Design for Signal Processing Using MATLAB and Mathematica. Prentice Hall.

NIST. Digital Library of Mathematical Functions. Retrieved December 3, 2020 from: https://dlmf.nist.gov/22.3#F1