Jacobi elliptic functions

They are found in the description of the motion of a pendulum, as well as in the design of electronic elliptic filters.

, the twelve functions form a repeating lattice of simple poles and zeroes.

[3] Depending on the function, one repeating parallelogram, or unit cell, will have sides of length

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)[note 1] In the above, the value

[6] 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

this way gives rise to very complicated branch cuts in the

and by analytic continuation in each of the variables otherwise: the Jacobi epsilon function is meromorphic in the whole complex plane (in both

arc length of the unit circle measured from the positive x-axis.

Similarly, Jacobi elliptic functions are defined on the unit ellipse,[citation needed] with a = 1.

Then the familiar relations from the unit circle: read for the ellipse: So the projections of the intersection point

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

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

Introducing complex numbers, our ellipse has an associated hyperbola: from applying Jacobi's imaginary transformation[11] to the elliptic functions in the above equation for x and y.

[13] 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.

The double periodicity of the Jacobi elliptic functions may be expressed as: where α and β are any pair of integers.

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 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.

Multiplying by any function of the form nq yields more general equations:

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[3]

The Jacobi epsilon and zn functions satisfy a quasi-addition theorem:

Double angle formulae can be easily derived from the above equations by setting x = y.

The derivatives of the three basic Jacobi elliptic functions (with respect to the first variable, with

These can be used to derive the derivatives of all other functions as shown in the table below (arguments (u,m) suppressed): Also 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: The function which exactly solves the pendulum differential equation, with initial angle

Bivariate power series expansions have been published by Schett.

There is an alternative method, based on the arithmetic-geometric mean and Landen's transformations:[6] Initialize where

[note 2] 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:[16] Let Set Then as

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

They can be represented as elliptic integrals,[23][24][25] and power series representations have been found.

The fundamental rectangle in the complex plane of
Model of the Jacobi amplitude (measured along vertical axis) as a function of independent variables u and the modulus k
Plot of the Jacobi ellipse ( x 2 + y 2 / b 2 = 1, b real) and the twelve Jacobi elliptic functions pq ( u , m ) for particular values of angle φ and parameter b . The solid curve is the ellipse, with m = 1 − 1/ b 2 and u = F ( φ , m ) where F (⋅,⋅) is the elliptic integral of the first kind (with parameter ). The dotted curve is the unit circle. Tangent lines from the circle and ellipse at x = cd crossing the x -axis at dc are shown in light grey.
The region in the complex plane. It is bounded by two semicircles from below, by a ray from the left and by a ray from the right.
Plot of the degenerate Jacobi curve ( x 2 + y 2 / b 2 = 1, b = ∞) and the twelve Jacobi Elliptic functions pq( u ,1) for a particular value of angle φ . The solid curve is the degenerate ellipse ( x 2 = 1) with m = 1 and u = F ( φ ,1) where F (⋅,⋅) is the elliptic integral of the first kind. The dotted curve is the unit circle. Since these are the Jacobi functions for m = 0 (circular trigonometric functions) but with imaginary arguments, they correspond to the six hyperbolic trigonometric functions.
Plot of the Jacobi hyperbola ( x 2 + y 2 / b 2 = 1, b imaginary) and the twelve Jacobi Elliptic functions pq( u , m ) for particular values of angle φ and parameter b . The solid curve is the hyperbola, with m = 1 − 1/ b 2 and u = F ( φ , m ) where F (⋅,⋅) is the elliptic integral of the first kind. The dotted curve is the unit circle. For the ds-dc triangle, σ =  sin( φ )cos( φ ).
Plots of the phase for the twelve Jacobi Elliptic functions pq(u,m) as a function complex argument u, with poles and zeroes indicated. The plots are over one full cycle in the real and imaginary directions with the colored portion indicating phase according to the color wheel at the lower right (which replaces the trivial dd function). Regions with absolute value below 1/3 are colored black, roughly indicating the location of a zero, while regions with absolute value above 3 are colored white, roughly indicating the position of a pole. All plots use m = 2/3 with K = K ( m ), K ′ = K (1 − m ), K (⋅) being the complete elliptic integral of the first kind. Arrows at the poles point in direction of zero phase. Right and left arrows imply positive and negative real residues respectively. Up and down arrows imply positive and negative imaginary residues respectively.