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 ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for . 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, p. 48 but his work was published much later.
Overview
There are twelve Jacobi elliptic functions denoted by , where and are any of the letters , , , and . (Functions of the form are trivially set to unity for notational completeness.) is the argument, and is the parameter, both of which may be complex. In fact, the Jacobi elliptic functions are meromorphic in both and . The distribution of the zeros and poles in the -plane is well-known. However, questions of the distribution of the zeros and poles in the -plane remain to be investigated.
In the complex plane of the argument , the twelve functions form a repeating lattice of simple poles and zeroes. Depending on the function, one repeating parallelogram, or unit cell, will have sides of length or on the real axis, and or on the imaginary axis, where and are known as the quarter periods with being the elliptic integral of the first kind. The nature of the unit cell can be determined by inspecting the "auxiliary rectangle" (generally a parallelogram), which is a rectangle formed by the origin at one corner, and as the diagonally opposite corner. As in the diagram, the four corners of the auxiliary rectangle are named , , , and , going counter-clockwise from the origin. The function will have a zero at the corner and a pole at the corner. The twelve functions correspond to the twelve ways of arranging these poles and zeroes in the corners of the rectangle.
When the argument and parameter are real, with , and 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.
Since the Jacobi elliptic functions are doubly periodic in , 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. The circumference of the first circle is and the second , where and 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: There is a simple zero at the corner , and a simple pole at the corner . The complex number is equal to half the period of the function ; that is, the function is periodic in the direction , with the period being . The function is also periodic in the other two directions and , with periods such that and 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 , or more commonly, in terms of given below. The second variable might be given in terms of the parameter , or as the elliptic modulus , where , or in terms of the modular angle , where . The complements of and are defined as and . These four terms are used below without comment to simplify various expressions.
The twelve Jacobi elliptic functions are generally written as where and are any of the letters , , , and . Functions of the form are trivially set to unity for notational completeness. The “major” functions are generally taken to be , and 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.)
Throughout this article, .
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 . These functions take the parameters and as inputs. The that satisfies
is called the Jacobi amplitude:
In this framework, the elliptic sine sn u (Latin: sinus amplitudinis) is given by
and the elliptic cosine cn u (Latin: cosinus amplitudinis) is given by
and the delta amplitude dn u (Latin: delta amplitudinis)If and is restricted to , then can be also written as
In the above, the value is a free parameter, usually taken to be real such that (but can be complex in general), and so the elliptic functions can be thought of as being given by two variables, and the parameter . The remaining nine elliptic functions are easily built from the above three (, , ), and are given in a section below. Note that when , that then equals the quarter period .
In the most general setting, is a multivalued function (in ) with infinitely many logarithmic branch points (the branches differ by integer multiples of ), namely the points and where . This multivalued function can be made single-valued by cutting the complex plane along the line segments joining these branch points (the cutting can be done in non-equivalent ways, giving non-equivalent single-valued functions), thus making analytic everywhere except on the branch cuts. In contrast, and other elliptic functions have no branch points, give consistent values for every branch of , and are meromorphic in the whole complex plane. Since every elliptic function is meromorphic in the whole complex plane (by definition), (when considered as a single-valued function) is not an elliptic function.
However, a particular cutting for can be made in the -plane by line segments from to with ; then it only remains to define at the branch cuts by continuity from some direction. Then becomes single-valued and singly-periodic in with the minimal period and it has singularities at the logarithmic branch points mentioned above. If and , is continuous in on the real line. When , the branch cuts of in the -plane cross the real line at for ; therefore for , is not continuous in on the real line and jumps by on the discontinuities.
But defining this way gives rise to very complicated branch cuts in the -plane (not the -plane); they have not been fully described as of yet.
Let
be the incomplete elliptic integral of the second kind with parameter .
Then the Jacobi epsilon function can be defined as
for and and by analytic continuation in each of the variables otherwise: the Jacobi epsilon function is meromorphic in the whole complex plane (in both and ). Alternatively, throughout both the -plane and -plane,
is well-defined in this way because all residues of are zero, so the integral is path-independent. So the Jacobi epsilon relates the incomplete elliptic integral of the first kind to the incomplete elliptic integral of the second kind:
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 function is defined by
It is a singly periodic function which is meromorphic in , but not in (due to the branch cuts of and ). Its minimal period in is . It is related to the Jacobi zeta function by
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.
In modern terms, the relation to elliptic integrals would be expressed by (or ) instead of .
Definition as trigonometry: the Jacobi ellipse
are defined on the unit circle with radius and angle arc length of the unit circle measured from the positive x-axis. Similarly, Jacobi elliptic functions are defined on the unit ellipse with and . Let
Then and
For each angle the parameter
(the incomplete elliptic integral of the first kind) is computed. On the unit circle (), would be an arc length. However, the relation of to the arc length of an ellipse is more complicated.
Let be a point on the ellipse, and let be the point where the unit circle intersects the line between and the origin . Then the familiar relations from the unit circle
read for the ellipse
So the projections of the intersection point of the line with the unit circle on the x- and y-axes are simply and . These projections may be interpreted as 'definition as trigonometry'. In short,
For the and values of the point with and parameter we get, by inserting the relation
into and , we find
The latter relations for the x- and y-coordinates of points on the unit ellipse may be considered as generalization of the relations and 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,dn) with . + Jacobi elliptic functions pq[u,m] as functions of {x,y,r} and {φ,dn}q c s n dp c1 s 1 n 1 d 1
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. With such that , let
and let , , . Then with , , and ,
The Jacobi zn function can be expressed by theta functions as well:
where 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
assumes every value in once and only once in
where is the upper half-plane in the complex plane, is the boundary of and
In this way, each can be associated with one and only one . Then Whittaker & Watson define the Jacobi elliptic functions by
where . In the book, they place an additional restriction on (that ), but it is in fact not a necessary restriction (see the Cox reference). Also, if or , 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:
Using the multiplication rule, all other functions may be expressed in terms of the above three. The transformations may be generally written as . The following table gives the for the specified pq(u,m). (The arguments are suppressed)