Jacobi's elliptic functions

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In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e.g. the equation of the pendulum—also see pendulum (mathematics)). They also have useful analogies to the functions of trigonometry, as indicated by the matching notation sn for sin. They are not the simplest way to develop a general theory, as now seen: that can be said for the Weierstrass elliptic functions. They are not, however, outmoded. They were introduced by Carl Gustav Jakob Jacobi, around 1830.

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[edit] Introduction

Auxiliary rectangle construction
Auxiliary rectangle construction

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. The rectangle is understood to be lying on the complex plane, so that s is at the origin, c is at the point K on the real axis, d is at the point K + iK' and n is at point iK' on the imaginary axis. The numbers K and K' are called the quarter periods. The twelve Jacobian elliptic functions are then pq, where each of p and q is 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:

  • There is a simple zero at the corner p, and a simple pole at the corner q.
  • The step from p to q is equal to half the period of the function pq u; that is, the function pq u is periodic in the direction pq, with the period being twice the distance from p to q. Also, pq u is also periodic in the other two directions as well, with a period such that the distance from p to one of the other corners is a quarter period.
  • If the function pq u is expanded in terms of u at one of the corners, the leading term in the expansion has a coefficient of 1. In other words, the leading term of the expansion of pq u at the corner p is u; the leading term of the expansion at the corner q is 1/u, and the leading term of an expansion at the other two corners is 1.

The Jacobian elliptic functions are then the unique elliptic functions that satisfy the above 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.

[edit] Notation

The elliptic functions can be given in a variety of notations, which can make the subject un-necessarily 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 o\!\varepsilon\,\!, where m=\sin^2o\!\varepsilon\,\!. 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.

[edit] 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. This is perhaps the easiest definition to understand. Let

u=\int_0^\phi \frac{d\theta} {\sqrt {1-m \sin^2 \theta}}.

Then the elliptic function sn u is given by

\operatorname {sn}\; u = \sin \phi\,

and cn u is given by

\operatorname {cn}\; u = \cos \phi

and

\operatorname {dn}\; u = \sqrt {1-m\sin^2 \phi}.\,

Here, the angle φ is called the amplitude. On occasion, \operatorname {dn}\; u = \Delta(u) is called the delta amplitude. In the above, the value m is a free parameter, usually taken to be real, 0\leq m \leq 1, and so the elliptic functions can be thought of as being given by two variables, the amplitude φ and the parameter m.

The remaining nine elliptic functions are easily built from the above three, and are given in a section below.

Note that when φ = π / 2, that u then equals the quarter period K.

[edit] Definition in terms of theta functions

Equivalently, Jacobi's elliptic functions can be defined in terms of his theta functions. If we abbreviate \vartheta(0;\tau) as \vartheta, and \vartheta_{01}(0;\tau), \vartheta_{10}(0;\tau), \vartheta_{11}(0;\tau) respectively as \vartheta_{01}, \vartheta_{10}, \vartheta_{11} (the theta constants) then the elliptic modulus k is k=({\vartheta_{10} \over \vartheta})^2. If we set u = \pi \vartheta^2 z, we have

\mbox{sn}(u; k) = -{\vartheta \vartheta_{11}(z;\tau) \over \vartheta_{10} \vartheta_{01}(z;\tau)}


\mbox{cn}(u; k) = {\vartheta_{01} \vartheta_{10}(z;\tau) \over \vartheta_{10} \vartheta_{01}(z;\tau)}


\mbox{dn}(u; k) = {\vartheta_{01} \vartheta(z;\tau) \over \vartheta \vartheta_{01}(z;\tau)}

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 k' = \sqrt{1-k^2}, the complementary modulus. As a function of τ it is

k'(\tau) = ({\vartheta_{01} \over \vartheta})^2.

Let us first define

\ell = {1 \over 2} {1-\sqrt{k'} \over 1+\sqrt{k'}} = {1 \over 2} {\vartheta - \vartheta_{01} \over \vartheta + \vartheta_{01}}.

Then define the nome q as q = exp(πiτ) and expand \ell as a power series in the nome q, we obtain

\ell = {q+q^9+q^{25}+ \cdots \over 1+2q^4+2q^{16}+ \cdots}.

Reversion of series now gives

q = \ell+2\ell^5+15\ell^9+150\ell^{13}+1707\ell^{17}+20910\ell^{21}+268616\ell^{25}+\cdots.

Since we may reduce to the case where the imaginary part of τ is greater than or equal to \sqrt{3}/2, we can assume the absolute value of q is less than or equal to \exp(-\pi \sqrt{3}/2); for values this small the above series converges very rapidly and easily allows us to find the appropriate value for q.

[edit] Minor functions

It is conventional to denote the reciprocals of the three functions above by reversing the order of the two letters of the function name:

\operatorname{ns}(u)=1/\operatorname{sn}(u)
\operatorname{nc}(u)=1/\operatorname{cn}(u)
\operatorname{nd}(u)=1/\operatorname{dn}(u)

The ratios of the three primary functions are denoted by the first letter of the numerator followed by the first letter of the denominator:

\operatorname{sc}(u)=\operatorname{sn}(u)/\operatorname{cn(u)}
\operatorname{sd}(u)=\operatorname{sn}(u)/\operatorname{dn(u)}
\operatorname{dc}(u)=\operatorname{dn}(u)/\operatorname{cn(u)}
\operatorname{ds}(u)=\operatorname{dn}(u)/\operatorname{sn(u)}
\operatorname{cs}(u)=\operatorname{cn}(u)/\operatorname{sn(u)}
\operatorname{cd}(u)=\operatorname{cn}(u)/\operatorname{dn(u)}

More compactly, we can write

\operatorname{pq}(u)=\frac{\operatorname{pr}(u)}{\operatorname{qr(u)}}

where each of p, q, and r is any of the letters s, c, d, n, with the understanding that ss = cc = dd = nn = 1.

[edit] Addition theorems

The functions satisfy the two algebraic relations

\operatorname{cn}^2 + \operatorname{sn}^2 = 1,\,
\operatorname{dn}^2 + k^2 \operatorname{sn}^2 = 1.\,

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

\operatorname{cn}(x+y) = {\operatorname{cn}(x)\;\operatorname{cn}(y) - \operatorname{sn}(x)\;\operatorname{sn}(y)\;\operatorname{dn}(x)\;\operatorname{dn}(y) \over {1 - k^2 \;\operatorname{sn}^2 (x) \;\operatorname{sn}^2 (y)}},


\operatorname{sn}(x+y) = {\operatorname{sn}(x)\;\operatorname{cn}(y)\;\operatorname{dn}(y) + \operatorname{sn}(y)\;\operatorname{cn}(x)\;\operatorname{dn}(x) \over {1 - k^2 \;\operatorname{sn}^2 (x)\; \operatorname{sn}^2 (y)}},


\operatorname{dn}(x+y) = {\operatorname{dn}(x)\;\operatorname{dn}(y) - k^2 \;\operatorname{sn}(x)\;\operatorname{sn}(y)\;\operatorname{cn}(x)\;\operatorname{cn}(y) \over {1 - k^2 \;\operatorname{sn}^2 (x)\; \operatorname{sn}^2 (y)}}.

[edit] Relations between squares of the functions

-\operatorname{dn}^2(u)+m_1= -m\;\operatorname{cn}^2(u) = m\;\operatorname{sn}^2(u)-m
-m_1\;\operatorname{nd}^2(u)+m_1= -mm_1\;\operatorname{sd}^2(u) = m\;\operatorname{cd}^2(u)-m
m_1\;\operatorname{sc}^2(u)+m_1= m_1\;\operatorname{nc}^2(u) = \operatorname{dc}^2(u)-m
\operatorname{cs}^2(u)+m_1=\operatorname{ds}^2(u)=\operatorname{ns}^2(u)-m

where m + m1 = 1 and m = k2.

Additional relations between squares can be obtained by noting that \operatorname{pq}^2 \cdot \operatorname{qp}^2 = 1 and that \operatorname{pq}=\operatorname{pr}/\operatorname{qr} where p, q, r are any of the letters s, c, d, n and ss = cc = dd = nn = 1.

[edit] Expansion in terms of the nome

Let the nome be q = exp( − πK' / K) and let the argument be v = πu / (2K). Then the functions have expansions as Lambert series

\operatorname{sn}(u)=\frac{2\pi}{K\sqrt{m}} \sum_{n=0}^\infty \frac{q^{n+1/2}}{1-q^{2n+1}} \sin (2n+1)v,
\operatorname{cn}(u)=\frac{2\pi}{K\sqrt{m}} \sum_{n=0}^\infty \frac{q^{n+1/2}}{1+q^{2n+1}} \cos (2n+1)v,
\operatorname{dn}(u)=\frac{\pi}{2K} + \frac{2\pi}{K} \sum_{n=1}^\infty \frac{q^{n}}{1+q^{2n}} \cos 2nv.

[edit] Jacobi's elliptic functions as solutions of nonlinear ordinary differential equations

The derivatives of the three basic Jacobian elliptic functions are:

\frac{\mathrm{d}}{\mathrm{d}z}\, \mathrm{sn}\,(z; k) = \mathrm{cn}\,(z;k)\, \mathrm{dn}\,(z;k),


\frac{\mathrm{d}}{\mathrm{d}z}\, \mathrm{cn}\,(z; k) = -\mathrm{sn}\,(z;k)\, \mathrm{dn}\,(z;k),


\frac{\mathrm{d}}{\mathrm{d}z}\, \mathrm{dn}\,(z; k) = - k^2 \mathrm{sn}\,(z;k)\, \mathrm{cn}\,(z;k).

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:

  • \mathrm{sn}\,(x;k) solves the differential equations \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1+k^2) y - 2 k^2 y^3 = 0, and \left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2 = (1-y^2) (1-k^2 y^2)
  • \mathrm{cn}\,(x;k) solves the differential equations \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1-2k^2) y + 2 k^2 y^3 = 0, and \left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2 = (1-y^2) (1-k^2 + k^2 y^2)
  • \mathrm{dn}\,(x;k) solves the differential equations \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} - (2 - k^2) y + 2 y^3 = 0, and \left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2 = (y^2 - 1) (1 - k^2 - y^2)

[edit] External links

[edit] References

  • Naum Illyich Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
  • E. T. Whittaker and G. N. Watson A Course of Modern Analysis, (1940, 1996) Cambridge University Press. ISBN 0-521-58807-3