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Approximating the Jacobian Elliptic Functions

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graph
Jacobian elliptic functions
Seiffert spirals
α
0.5
number of periods
1
2
3
4
5
approximation order
1
2
3
4
5
6
7
8
9
10
sn
(α,ϕ)
cn
(α,ϕ)
dn
(α,ϕ)
Jacobian elliptic functions are extensions and generalizations of trigonometric sine and cosine functions. Applications in physics abound. For example, the functions
sn
and
cn
occur in the exact solution of the equations of motion for the plane pendulum [1]. Any approximation procedure to solve the pendulum equations of motion involves approximations to the Jacobian elliptic functions [2]. The approximation also pertains to Seiffert spirals [3], trajectories along the surface of a sphere.
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