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Jacobi Elliptic Functions

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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.

jacobi elliptic functions

Line graphs of the Jacobi elliptic functions nd(x, k), dn (x, k), cn (x, k) and sn (x, k), illustrating near trigonometric behavior. Image: NIST.



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:

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 = sin2 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) = difference function

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):

  • sn2(u, k) + cn(u, k) = 1
  • k2sn2(u, k) + dn2(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

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Stephanie Glen. "Jacobi Elliptic Functions" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/jacobi-elliptic-functions/
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