- Jacobi's elliptic functions
In
mathematics , the Jacobi elliptic functions are a set of basicelliptic function s, and auxiliarytheta function s, 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 thependulum —also seependulum (mathematics) ). They also have useful analogies to the functions oftrigonometry , 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 theWeierstrass elliptic functions . They are not, however, outmoded. They were introduced byCarl Gustav Jakob Jacobi , around 1830.Introduction
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 thequarter period s. 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.
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 "k"2 = "m", or in terms of themodular 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 onelliptic integrals andquarter 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. This is perhaps the easiest definition to understand. Let:u=int_0^phi frac{d heta} {sqrt {1-m sin^2 heta.
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-msin^2 phi}.,
Here, the angle phi 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, 0leq m leq 1, and so the elliptic functions can be thought of as being given by two variables, the amplitude phi 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 phi=pi/2, that "u" then equals the
quarter period "K".Definition in terms of theta functions
Equivalently, Jacobi's elliptic functions can be defined in terms of his
theta function s. If we abbreviate vartheta(0; au) as vartheta, and vartheta_{01}(0; au), vartheta_{10}(0; au), vartheta_{11}(0; au) respectively as vartheta_{01}, vartheta_{10}, vartheta_{11} (the "theta constants") then theelliptic 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; au) over vartheta_{10} vartheta_{01}(z; au)}
:mbox{cn}(u; k) = {vartheta_{01} vartheta_{10}(z; au) over vartheta_{10} vartheta_{01}(z; au)}
:mbox{dn}(u; k) = {vartheta_{01} vartheta(z; au) over vartheta vartheta_{01}(z; au)}
Since the Jacobi functions are defined in terms of the elliptic modulus k( au), 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'( au) = ({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 (pi i au) 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+2ell^5+15ell^9+150ell^{13}+1707ell^{17}+20910ell^{21}+268616ell^{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".
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.
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 twoquadric s 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).
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+m_1=1 and m=k^2.
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.
Expansion in terms of the nome
Let the nome be q=exp(-pi K'/K) and let the argument be v=pi u /(2K). Then the functions have expansions as
Lambert series :operatorname{sn}(u)=frac{2pi}{Ksqrt{msum_{n=0}^infty frac{q^{n+1/2{1-q^{2n+1 sin (2n+1)v,
:operatorname{cn}(u)=frac{2pi}{Ksqrt{msum_{n=0}^infty frac{q^{n+1/2{1+q^{2n+1 cos (2n+1)v,
:operatorname{dn}(u)=frac{pi}{2K} + frac{2pi}{K}sum_{n=1}^infty frac{q^{n{1+q^{2n cos 2nv.
Jacobi's elliptic functions as solutions of nonlinear ordinary differential equations
The
derivative s 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 equation s:* 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} ight)^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} ight)^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} ight)^2 = (y^2 - 1) (1 - k^2 - y^2)
External links
* [http://mathworld.wolfram.com/JacobiEllipticFunctions.html Eric W. Weisstein, "Jacobi Elliptic Functions" (Mathworld)]
References
*cite book
last = Abramowitz
first = Milton
authorlink =
coauthors = Stegun, Irene A. eds.
title =Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
accessdate = 2007-07-25
date = 1972
publisher = Dover
location = New York
id = ISBN 0-486-61272-4 See [http://www.math.sfu.ca/~cbm/aands/page_569.htm Chapter 16]* 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*
Alfred George Greenhill [http://www.archive.org/details/applicationselli00greerich The applications of elliptic functions] (London, New York, Macmillan, 1892)
* H. Hancock [http://www.archive.org/details/lecturestheorell00hancrich Lectures on the theory of elliptic functions] (New York, J. Wiley & sons, 1910)
* A. C. Dixon [http://www.archive.org/details/117736039 The elementary properties of the elliptic functions, with examples] (Macmillan, 1894)
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