Approximating the Jacobian Elliptic Functions

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Jacobian elliptic functions are extensions and generalizations of trigonometric sine and cosine functions. Applications in physics abound. For example, the functions and 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.

Contributed by: Brad Klee (June 2016)
Open content licensed under CC BY-NC-SA



The axis of the graphs for , , and is measured in units of , the complete elliptic integral of the first kind. The functions , have period , while the function has period . The term-by-term expansion of is given [4] by


Approximations of the Jacobian elliptic functions , , and in parametric form follow from an arbitrary-precision solution of the plane pendulum's equations of motion [1, 3]:





where are the polar coordinates of phase space. The function gives the relation between the angle and the domain of the Jacobian elliptic functions, . The approximate phase radius and time dependence are:



where the coefficients and are computable rational numbers with closed form as yet unknown (Cf. OEIS A273506, A274130 [5]).

In the complete limit, as the approximation is exact. Choosing some finite introduces error at order . One easy way to see the error is to consider the approximate period, which is directly proportional to the dependent factor:


In the plots, vertical lines show convergence toward the exact period as increases; however, cumulative errors always become evident if the function domain or time chosen is large enough. Convergence depends strongly on the chosen value of .

For definitions of the Seiffert spirals see the wonderful article by Erdös [3].


[1] A. J. Brizard, "A Primer on Elliptic Functions with Applications in Classical Mechanics."

[2] B. Klee, "Plane Pendulum and Beyond by Phase Space Geometry."

[3] P. Erdös, "Spiraling the Earth with C. G. J. Jacobi," American Journal of Physics 68, 2000 pp. 888–895. doi:10.1119/1.1285882.

[4] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.

[5] The OEIS Foundation. The Online Encyclopedia of Integer Sequences. A273506, A273507, A274130, A274131, A274076,

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