10
Jacobi Elliptic Functions
Abstract
Topic of this chapter are the Jacobi elliptic functions sn, cn, dn, their first
derivatives, the additional nine Jacobi elliptic functions, and the amplitude
function am. The evaluation of the Jacobi elliptic functions is based on the
Jacobi ϑ functions and in case of convergency issues additionally on the Gauss
hypergeometric function. The corresponding m-files and some additional utility
functions are available for download.
Jacobi elliptic functions are functions that appear in a variety of applications in
engineering and physics, e.g., in hydrodynamics, general relativity, and classical
dynamics (see example: motion in a quartic potential), to list only a few.
The elliptic integral of first kind occurs, e.g., in the solution of the pendulum. Its
Legendre and its trigonometric form is given by
F (φ, k) =
sin φ
0
1
(1 − t 2 )(1 − k 2 t 2 )
dt
(10.1a)
=
φ
0
1
1 − k 2 sin
2 θ
dθ
(10.1b)
with 0 ≤ k 2 ≤ 1 and 0 ≤ φ ≤ π/2. With respect to the notation, we follow [2, 3],
whereas Abramowitz et al. [1] are using m = k 2 .
The complete integral of first kind is given by
K(k) =
π/2
0
1
1 − k 2 sin
2 θ
dθ
(10.1c)
= F
π
2
, k
.
(10.1d)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_10
131