### 16.1 Introduction to Elliptic Functions and Integrals

Maxima includes support for Jacobian elliptic functions and for complete and incomplete elliptic integrals. This includes symbolic manipulation of these functions and numerical evaluation as well. Definitions of these functions and many of their properties can by found in Abramowitz and Stegun, A&S Chapter 16 and A&S Chapter 17. See also DLMF 22.2. As much as possible, we use the definitions and relationships given in Abramowitz and Stegun.

In particular, all elliptic functions and integrals use the parameter m instead of the modulus k or the modular angle \alpha. The following relationships are true:

$$ \eqalign{ m &= k^2 \cr k &= \sin\alpha } $$

Note that Abramowitz and Stegun uses the notation \({\rm sn}(u|m)\) where we use \({\rm sn}(u,m)\) instead. The DLMF uses modulus k instead of the parameter m.

The elliptic functions and integrals are primarily intended to support symbolic computation. Therefore, most of derivatives of the functions and integrals are known. However, if floating-point values are given, a floating-point result is returned.

Support for most of the other properties of elliptic functions and integrals other than derivatives has not yet been written.

Some examples of elliptic functions:

    (%i1) jacobi_sn (u, m);
    (%o1)                    jacobi_sn(u, m)
    (%i2) jacobi_sn (u, 1);
    (%o2)                        tanh(u)
    (%i3) jacobi_sn (u, 0);
    (%o3)                        sin(u)
    (%i4) diff (jacobi_sn (u, m), u);
    (%o4)            jacobi_cn(u, m) jacobi_dn(u, m)
    (%i5) diff (jacobi_sn (u, m), m);
    (%o5) jacobi_cn(u, m) jacobi_dn(u, m)

          elliptic_e(asin(jacobi_sn(u, m)), m)
     (u - ------------------------------------)/(2 m)
                         1 - m

                2
       jacobi_cn (u, m) jacobi_sn(u, m)
     + --------------------------------
                  2 (1 - m)

Some examples of elliptic integrals:

    (%i1) elliptic_f (phi, m);
    (%o1)                  elliptic_f(phi, m)
    (%i2) elliptic_f (phi, 0);
    (%o2)                          phi
    (%i3) elliptic_f (phi, 1);
                                   phi   %pi
    (%o3)                  log(tan(--- + ---))
                                    2     4
    (%i4) elliptic_e (phi, 1);
    (%o4)                       sin(phi)
    (%i5) elliptic_e (phi, 0);
    (%o5)                          phi
    (%i6) elliptic_kc (1/2);
                                         1
    (%o6)                    elliptic_kc(-)
                                         2
    (%i7) makegamma (%);
                                     2 1
                                gamma (-)
                                       4
    (%o7)                      -----------
                               4 sqrt(%pi)
    (%i8) diff (elliptic_f (phi, m), phi);
                                    1
    (%o8)                 ---------------------
                                        2
                          sqrt(1 - m sin (phi))
    (%i9) diff (elliptic_f (phi, m), m);
           elliptic_e(phi, m) - (1 - m) elliptic_f(phi, m)
    (%o9) (-----------------------------------------------
                                  m

                                     cos(phi) sin(phi)
                                 - ---------------------)/(2 (1 - m))
                                                 2
                                   sqrt(1 - m sin (phi))

Support for elliptic functions and integrals was written by Raymond Toy. It is placed under the terms of the General Public License (GPL) that governs the distribution of Maxima.

Categories:Elliptic functions ·