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In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that are of historical importance. Many of their features show up in important structures and have direct relevance to some applications (e.g. the equation of a pendulum—also see pendulum (mathematics)). They also have useful analogies to the functions of trigonometry, as indicated by the matching notation sn for sin. The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by Carl Gustav Jakob Jacobi (1829).
Contents
- Introduction
- Notation
- Definition as inverses of elliptic integrals
- Definition as trigonometry
- Definition in terms of theta functions
- Minor functions
- Addition theorems
- Relations between squares of the functions
- Expansion in terms of the nome
- Jacobi elliptic functions as solutions of nonlinear ordinary differential equations
- Approximation in terms of hyperbolic functions
- Inverse functions
- Map projection
- References
Introduction
Jacobian elliptic functions are doubly periodic meromorphic functions on the complex plane. Since they are doubly periodic, they factor through a torus – in effect, their domain can be taken to be a torus, just as cosine and sine are in effect defined on a circle. Instead of having only one circle, we now have the product of two circles, one real and the other imaginary. The complex plane can be replaced by a complex torus. The circumference of the first circle is 4K and the second 4K′, where K and K′ are the quarter periods. Each function has two zeroes and two poles at opposite positions on the torus. Among the points 0, K, K + iK′, iK′ there is one zero and one pole. So an arrow can be drawn from a zero to a pole.
So there are twelve Jacobian elliptic functions. Each of the twelve corresponds to an arrow drawn from one corner of a rectangle to another. The corners of the rectangle are labeled, by convention, s, c, d and n. s is at the origin, c is at the point K on the real axis/loop, d is at the point K + iK′ and n is at point iK′ on the imaginary axis/loop. The twelve Jacobian elliptic functions are then pq, where each of p and q is a different one of the letters s, c, d, n.
The Jacobian elliptic functions are then the unique doubly periodic, meromorphic functions satisfying the following three properties:
More generally, there is no need to impose a rectangle; a parallelogram will do. However, if K and iK' are kept on the real and imaginary axis respectively, then the Jacobi elliptic functions pq u will be real functions when u is real.
Notation
The elliptic functions can be given in a variety of notations, which can make the subject unnecessarily confusing. Elliptic functions are functions of two variables. The first variable might be given in terms of the amplitude φ, or more commonly, in terms of u given below. The second variable might be given in terms of the parameter m, or as the elliptic modulus k, where k2 = m, or in terms of the modular angle α, where m = sin2 α. A more extensive review and definition of these alternatives, their complements, and the associated notation schemes are given in the articles on elliptic integrals and quarter period.
Definition as inverses of elliptic integrals
The above definition, in terms of the unique meromorphic functions satisfying certain properties, is quite abstract. There is a simpler, but completely equivalent definition, giving the elliptic functions as inverses of the incomplete elliptic integral of the first kind. Let
Then the elliptic function sn u is given by
and cn u is given by
and
Here, the angle
The remaining nine elliptic functions are easily built from the above three, and are given in a section below.
Note that when
Definition as trigonometry
then:
For each angle
Let
imply for the ellipse:
So the projections of the intersection point
For the
into:
The latter relations for the x- and y-coordinates of points on the unit ellipse may be considered as generalization of the relations
Introducing complex numbers, our ellipse has a dual:
from applying Jacobi's imaginary transformation to the elliptic functions in the above equation for x and y.
It follows that we can put
Definition in terms of theta functions
Equivalently, Jacobi's elliptic functions can be defined in terms of his theta functions. If we abbreviate
Since the Jacobi functions are defined in terms of the elliptic modulus k(τ), we need to invert this and find τ in terms of k. We start from
Let us first define
Then define the nome q as
Reversion of series now gives
Since we may reduce to the case where the imaginary part of τ is greater than or equal to 1/2 sqrt(3), we can assume the absolute value of q is less than or equal to exp(-1/2 sqrt(3) π) ~ 0.0658; for values this small the above series converges very rapidly and easily allows us to find the appropriate value for q.
Minor functions
Reversing the order of the two letters of the function name results in the reciprocals of the three functions above:
Similarly, the ratios of the three primary functions correspond to the first letter of the numerator followed by the first letter of the denominator:
More compactly, we have
where p and q are any of the letters s, c, d.
(This notation is due to Gudermann and Glaisher and is not Jacobi's original notation
Addition theorems
The functions satisfy the two algebraic relations
From this we see that (cn, sn, dn) parametrizes an elliptic curve which is the intersection of the two quadrics defined by the above two equations. We now may define a group law for points on this curve by the addition formulas for the Jacobi functions
Relations between squares of the functions
where m + m1 = 1 and m = k2.
Additional relations between squares can be obtained by noting that pq2 · qp2 = 1 and that pq = pr / qr where p, q, r are any of the letters s, c, d, n and ss = cc = dd = nn = 1.
Expansion in terms of the nome
Let the nome be
Jacobi elliptic functions as solutions of nonlinear ordinary differential equations
The derivatives of the three basic Jacobi elliptic functions are:
With the addition theorems above and for a given k with 0 < k < 1 they therefore are solutions to the following nonlinear ordinary differential equations:
Approximation in terms of hyperbolic functions
The Jacobi elliptic functions can be expanded in terms of the hyperbolic functions. When
Inverse functions
The inverses of the Jacobi elliptic functions can be defined similarly to the inverse trigonometric functions; if
Map projection
The Peirce quincuncial projection is a map projection based on Jacobian elliptic functions.