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Wikipedia / Jacobi elliptic functions

In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections. 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 Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later.

Overview

is the parameter, both of which may be complex. In fact, the Jacobi elliptic functions are meromorphic.-plane remain to be investigated.

In the complex plane, the twelve functions form a repeating lattice of simple poles and zeroes. Depending on the function, one repeating parallelogram are known as the quarter periods being the elliptic integral corner. The twelve functions correspond to the twelve ways of arranging these poles and zeroes in the corners of the rectangle.

will be real and the auxiliary parallelogram will in fact be a rectangle, and the Jacobi elliptic functions will all be real valued on the real line.

, 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 are the quarter periods. Each function has two zeroes and two poles at opposite positions on the torus. Among the points there is one zero and one pole.

The Jacobi elliptic functions are then doubly periodic, meromorphic functions satisfying the following properties:

  • , and a simple pole at the corner.

  • are quarter periods.

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 given below. The second variable might be given in terms of the parameter, or as the elliptic modulus, or in terms of the modular angle. These four terms are used below without comment to simplify various expressions.

from which all other functions can be derived and expressions are often written solely in terms of these three functions, however, various symmetries and generalizations are often most conveniently expressed using the full set. (This notation is due to Gudermann and Glaisher and is not Jacobi's original notation.)

.

The functions are notationally related to each other by the multiplication rule: (arguments suppressed)

from which other commonly used relationships can be derived:

The multiplication rule follows immediately from the identification of the elliptic functions with the Neville theta functions

Also note that:

Definition in terms of inverses of elliptic integrals

There is a definition, relating the elliptic functions to the inverse of the incomplete elliptic integral of the first kind that satisfies

is called the Jacobi amplitude:

In this framework, the elliptic sine snu (Latin: sinus amplitudinis) is given by

and the elliptic cosine cnu (Latin: cosinus amplitudinis) is given by

and the delta amplitude dnu (Latin: delta amplitudinis)

is a free parameter and the parameter then equals the quarter period.

is a multivalued function) with infinitely many logarithmic branch points. analytic everywhere except on the branch cuts, and are meromorphic (when considered as a single-valued function) is not an elliptic function.

on the discontinuities.

-plane (not-plane); they have not been fully described as of yet.

be the incomplete elliptic integral of the second kind.

Then the Jacobi epsilon and by analytic continuation-plane, is well-defined in this way because all residues The Jacobi epsilon function is not an elliptic function, but it appears when differentiating the Jacobi elliptic functions with respect to the parameter.

The Jacobi zn It is a singly periodic function. It is related to the Jacobi zeta function

Historically, the Jacobi elliptic functions were first defined by using the amplitude. In more modern texts on elliptic functions, the Jacobi elliptic functions are defined by other means, for example by ratios of theta functions (see below), and the amplitude is ignored.

.

Definition as trigonometry: the Jacobi ellipse

are defined on the unit circle arc length of the unit circle measured from the positive x-axis. Similarly, Jacobi elliptic functions are defined on the unit ellipse. Let

and

to the arc length of an ellipse is more complicated.

. Then the familiar relations from the unit circle

read for the ellipse

with the unit circle on the x- and y. These projections may be interpreted as 'definition as trigonometry'. In short,

we get, by inserting the relation

, we find

The latter relations for the x- and y for the coordinates of points on the unit circle.

The following table summarizes the expressions for all Jacobi elliptic functions pq(u,m) in the variables (x,y,r) and (φ.

Definition in terms of the Jacobi theta functions

Using elliptic integrals

Equivalently, Jacobi's elliptic functions can be defined in terms of the theta functions., let

,

denotes the partial derivative with respect to the first variable.

Using modular inversion

In fact, the definition of the Jacobi elliptic functions in Whittaker & Watson is stated a little bit differently than the one given above (but it's equivalent to it) and relies on modular inversion: The function, defined by once and only once is the upper half-plane can be associated with one and only one, the Jacobi elliptic functions degenerate to non-elliptic functions which is described below.

Definition in terms of Neville theta functions

The Jacobi elliptic functions can be defined very simply using the Neville theta functions:

Simplifications of complicated products of the Jacobi elliptic functions are often made easier using these identities.

Jacobi transformations

The Jacobi imaginary transformations

The Jacobi imaginary transformations relate various functions of the imaginary variable i u or, equivalently, relations between various values of the m parameter. In terms of the major functions:

for the specified pq(u,m). are suppressed)Since the hyperbolic trigonometric functions are proportional to the circular trigonometric functions with imaginary arguments, it follows that the Jacobi functions will yield the hyperbolic functions for m=1. In the figure, the Jacobi curve has degenerated to two vertical lines at x=1 and x=−1.

The Jacobi real transformations

The Jacobi real transformations yield expressions for the elliptic functions in terms with alternate values of m for the specified pq(u,m). are suppressed)

Other Jacobi transformations

Jacobi's real and imaginary transformations can be combined in various ways to yield three more simple transformations. The real and imaginary transformations are two transformations in a group (D3 or anharmonic group) of six transformations. If

is the transformation for the m parameter in the real transformation, and

is the transformation of m in the imaginary transformation, then the other transformations can be built up by successive application of these two basic transformations, yielding only three more possibilities:

These five transformations, along with the identity transformation (U(m)=m) yield the six-element group. With regard to the Jacobi elliptic functions, the general transformation can be expressed using just three functions:

where i = U, I, IR, R, RI, or RIR, identifying the transformation, i is a multiplication factor common to these three functions, and the prime indicates the transformed function. The other nine transformed functions can be built up from the above three. The reason the cs, ns, ds functions were chosen to represent the transformation is that the other functions will be ratios of these three (except for their inverses) and the multiplication factors will cancel.

The following table lists the multiplication factors for the three ps functions, the transformed ms, and the transformed function names for each of the six transformations. (As usual, k2=m, 1−k2=k12=m) are suppressed)Thus, for example, we may build the following table for the RIR transformation. are suppressed)The value of the Jacobi transformations is that any set of Jacobi elliptic functions with any real-valued parameter m and, for real values of u, the function values will be real.

Amplitude transformations

:

such that both sides are well-defined.

With

we have

such that both sides are well-defined.

The Jacobi hyperbola

Introducing complex numbers, our ellipse has an associated hyperbola:

from applying Jacobi's imaginary transformation to the elliptic functions in the above equation for x andy.

. So our ellipse has a dual ellipse with m replaced by 1-m. This leads to the complex torus mentioned in the Introduction. Generally, m may be a complex number, but when m is real and m<0, the curve is an ellipse with major axis in the x direction. At m=0 the curve is a circle, and for 0<m<1, the curve is an ellipse with major axis in the y direction. At m=1, the curve degenerates into two vertical lines at x=±1. For m>1, the curve is a hyperbola. When m is complex but not real, x or y or both are complex and the curve cannot be described on a real x-y diagram.

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.

Periodicity, poles, and residues

In the complex plane of the argument u, the Jacobi elliptic functions form a repeating pattern of poles (and zeroes). The residues of the poles all have the same absolute value, differing only in sign. Each function pq(u,m) has an "inverse function" (in the multiplicative sense) qp(u,m) in which the positions of the poles and zeroes are exchanged. The periods of repetition are generally different in the real and imaginary directions, hence the use of the term "doubly periodic" to describe them.

is the complete elliptic integral of the second kind.

The double periodicity of the Jacobi elliptic functions may be expressed as:

where and are any pair of integers. K() is the complete elliptic integral of the first kind, also known as the quarter period. The power of negative unity () is given in the following table:When the factor (−1) is equal to −1, the equation expresses quasi-periodicity. When it is equal to unity, it expresses full periodicity. It can be seen, for example, that for the entries containing only when is even, full periodicity is expressed by the above equation, and the function has full periods of 4K(m) and 2iK(1−m). Likewise, functions with entries containing only have full periods of 2K(m) and 4iK(1−m), while those with + have full periods of 4K(m) and 4iK(1−m).

In the diagram on the right, which plots one repeating unit for each function, indicating phase along with the location of poles and zeroes, a number of regularities can be noted: The inverse of each function is opposite the diagonal, and has the same size unit cell, with poles and zeroes exchanged. The pole and zero arrangement in the auxiliary rectangle formed by (0,0), (K,0), (0,K) and (K,K) are in accordance with the description of the pole and zero placement described in the introduction above. Also, the size of the white ovals indicating poles are a rough measure of the absolute value of the residue for that pole. The residues of the poles closest to the origin in the figure (i.e. in the auxiliary rectangle) are listed in the following table:When applicable, poles displaced above by 2K or displaced to the right by 2K have the same value but with signs reversed, while those diagonally opposite have the same value. Note that poles and zeroes on the left and lower edges are considered part of the unit cell, while those on the upper and right edges are not.

The information about poles can in fact be used to characterize the Jacobi elliptic functions:

.

.

.

Special values

gives the lemniscate elliptic functions:

, the Jacobi elliptic functions are reduced to non-elliptic functions: is the Gudermannian function.

.

Identities

Half angle formula

K formulas

Half K formula

Third K formula

To get x3, we take the tangent of twice the arctangent of the modulus.

Also this equation leads to the sn-value of the third of K:

These equations lead to the other values of the Jacobi-Functions:

Fifth K formula

Following equation has following solution:

To get the sn-values, we put the solution x into following expressions:

Relations between squares of the functions

Relations between squares of the functions can be derived from two basic relationships (Arguments (u,m where m+m'=1. Multiplying by any function of the form nq yields more general equations:

With q=d), with x=cd, y=sd and r=nd. Using the multiplication rule, other relationships may be derived. For example:

Addition theorems

The functions satisfy the two square relations (dependence on m

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

Double angle formulae can be easily derived from the above equations by setting x=y. Half angle formulae are all of the form:

Jacobi elliptic functions as solutions of nonlinear ordinary differential equations

Derivatives with respect to the first variable

The derivatives

These can be used to derive the derivatives of all other functions as shown in the table below (arguments (u,m) suppressed):

With the addition theorems above and for a given m with 0<m<1 the major functions are therefore solutions to the following nonlinear ordinary differential equations:

  • and

not on a branch cut)

The function which exactly solves the pendulum differential equation and zero initial angular velocity is

.

Derivatives with respect to the second variable

are as follows:

Expansion in terms of the nome

Let the nome. Then the functions have expansions as Lambert series

Bivariate power series expansions have been published by Schett.

Fast computation

The theta function ratios provide an efficient way of computing the Jacobi elliptic functions. There is an alternative method, based on the arithmetic-geometric mean and Landen's transformations:

on the real line.

In conjunction with the addition theorems for elliptic functions (which hold for complex numbers in general) and the Jacobi transformations, the method of computation described above can be used to compute all Jacobi elliptic functions in the whole complex plane.

Another method of fast computation of the Jacobi elliptic functions via the arithmetic–geometric mean, avoiding the computation of the Jacobi amplitude, is due to Herbert E. Salzer:

.

Yet, another method for a rapidly converging fast computation of the Jacobi elliptic sine function found in the literature is shown below.

Let:

Then set:

Then:

.

Approximation in terms of hyperbolic functions

can be neglected, we have:

  • sn(u

  • cn(u

  • dn(u

Continued fractions

and the nome with elliptic modulus is the complete elliptic integral of the first kind, then holds the following continued fraction expansion are

:

:

:

:

:

Inverse functions

The inverses of the Jacobi elliptic functions can be defined similarly to the inverse trigonometric functions. They can be represented as elliptic integrals, and power series representations have been found.

Map projection

The Peirce quincuncial projection is a map projection based on Jacobian elliptic functions.

See also

Notes

Citations

References