Jacobi's elliptic functions - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Jacobi elliptic functions are a set of basic elliptic function

Elliptic function

In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...

s, 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

Pendulum

A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...

—also see pendulum (mathematics)

Pendulum (mathematics)

The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allows the equations of motion to be solved analytically for small-angle oscillations.- Simple gravity pendulum :...

). They also have useful analogies to the functions of trigonometry

Trigonometry

Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

, as indicated by the matching notation sn for sin. The Jacobi elliptic functions occur more in practical problems than the Weierstrass elliptic functions. They were introduced by Carl Gustav Jakob Jacobi

Carl Gustav Jakob Jacobi

Carl Gustav Jacob Jacobi was a German mathematician, widely considered to be the most inspiring teacher of his time and is considered one of the greatest mathematicians of his generation.-Biography:...

, 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

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

, 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. The function pq u is also periodic in the other two directions, 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.

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

, where

. 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

Elliptic integral

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

of the first kind. This is perhaps the easiest definition to understand. Let

Then the elliptic function sn u is given by

and cn u is given by

and

Here, the angle

is called the amplitude. On occasion, dn u = Δ(u) is called the delta amplitude. In the above, the value m is a free parameter, usually taken to be real, 0 ≤ m ≤ 1, and so the elliptic functions can be thought of as being given by two variables, the amplitude

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

, that u then equals the quarter period K.

Definition in terms of theta functions

Equivalently, Jacobi elliptic functions can be defined in terms of his theta functions. If we abbreviate

, and

respectively as

(the theta constants) then the elliptic modulus k is

. If we set

, we have

Since the Jacobi functions are defined in terms of the elliptic modulus

, we need to invert this and find τ in terms of k. We start from

, the complementary modulus. As a function of τ it is

Let us first define

Then define the nome q as

and expand

as a power series in the nome q, we obtain

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

(This notation is due to Gudermann and Glaisher

James Whitbread Lee Glaisher

James Whitbread Lee Glaisher son of James Glaisher, the meteorologist, was a prolific English mathematician.He was educated at St Paul's School and Trinity College, Cambridge, where he was second wrangler in 1871...

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

Elliptic curve

In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

which is the intersection of the two quadric

Quadric

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...

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

Relations between squares of the functions

where

and

.

Additional relations between squares can be obtained by noting that

and that

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

and let the argument be

. Then the functions have expansions as Lambert series

Jacobi elliptic functions as solutions of nonlinear ordinary differential equations

The derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

s 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 equation

Ordinary differential equation

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

solves the differential equations

and

solves the differential equations

and

solves the differential equations

and

Map projection

The Peirce quincuncial projection

Peirce quincuncial projection

The Peirce quincuncial projection is a conformal map projection that presents the sphere as a square...

is a map projection

Map projection

A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...

based on Jacobian elliptic functions.

Introduction

Notation

Definition as inverses of elliptic integrals

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

Map projection

See also