NationMaster - Encyclopedia: Jacobi's elliptic functions

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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ... In mathematics, theta functions are special functions of several complex variables. ... For other uses, see Pendulum (disambiguation). ... The mathematics of pendulums can be quite complex, but some formula and proofs are given below. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigÅnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. ... Karl Gustav Jacob Jacobi Carl Gustav Jacob Jacobi (December 10, 1804 - February 18, 1851), was not only a great German mathematician but also considered by many as the most inspiring teacher of his time (Bell, p. ...

1 Introduction
2 Notation
3 Definition as inverses of elliptic integrals
4 Definition in terms of theta functions
5 Minor functions
6 Addition theorems
7 Relations between squares of the functions
8 Expansion in terms of the nome
9 Jacobi's elliptic functions as solutions of nonlinear ordinary differential equations
10 External links
11 References

Introduction

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. Image File history File links JacobiFunctionAbstract. ... Image File history File links JacobiFunctionAbstract. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In mathematics, the quarter periods K(m) and iK′(m) are special functions that appear in the theory of elliptic functions. ...

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 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. In the study of ellipses and related geometry, various parameters in the distortion of a circle into an ellipse are identified and employed: Aspect ratio, flattening and eccentricity. ... In integral calculus, an elliptic integral is any function f which can be expressed in the form where R is a rational function of its two arguments, P is the square root of a polynomial of degree 3 or 4 (a cubic or quartic) with no repeated roots, and c... In mathematics, the quarter periods K(m) and iK′(m) are special functions that appear in the theory of elliptic functions. ...

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 In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ...

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

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

$operatorname {dn}; u = sqrt {1-msin^2 phi}.,$

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, $0leq m leq 1$ , 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. In mathematics, the quarter periods K(m) and iK′(m) are special functions that appear in the theory of elliptic functions. ...

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 In mathematics, theta functions are special functions of several complex variables. ...

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

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

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

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 $q = exp(π i τ)$ and expand $ell$ as a power series in the nome q, we obtain In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by where K and iK are the quarter periods, and and are the fundamental pair of periods. ...

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

Reversion of series now gives In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange-Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. ...

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

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 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 In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ...

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

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

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

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 $q = exp( - π K' / K)$ and let the argument be $v = π u / (2 K)$ . Then the functions have expansions as Lambert series In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by where K and iK are the quarter periods, and and are the fundamental pair of periods. ... In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form It can be resummed formally by expanding the denominator: where the coefficients of the new series are given by the Dirichlet convolution of with the constant function : This series may be inverted by means...

$operatorname{sn}(u)=frac{2pi}{Ksqrt{m}} sum_{n=0}^infty frac{q^{n+1/2}}{1-q^{2n+1}} sin (2n+1)v,$

$operatorname{cn}(u)=frac{2pi}{Ksqrt{m}} sum_{n=0}^infty frac{q^{n+1/2}}{1+q^{2n+1}} cos (2n+1)v,$

$operatorname{dn}(u)=frac{pi}{2K} + frac{2pi}{K} sum_{n=0}^infty frac{q^{n}}{1+q^{2n}} cos 2nv.$

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

The derivatives of the three basic Jacobian elliptic functions are: For a non-technical overview of the subject, see Calculus. ...

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

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

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

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: In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

$mathrm{sn},(x;k)$ 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 $left(frac{mathrm{d} y}{mathrm{d}x}right)^2 = (y^2 - 1) (1 - k^2 - y^2)$

External links

Eric W. Weisstein, "Jacobi Elliptic Functions" (Mathworld)

References

Abramowitz, Milton; Stegun, Irene A. eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4. See 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 The applications of elliptic functions (London, New York, Macmillan, 1892)
H. Hancock Lectures on the theory of elliptic functions (New York, J. Wiley & sons, 1910)
A. C. Dixon The elementary properties of the elliptic functions, with examples (Macmillan, 1894)

Category: Elliptic functions Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ... Sir Alfred George Greenhill, MA, FRS (November 29, 1847, London â€” February 10, 1927, London) was a British mathematician. ...

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