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:
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.
See A&S Section 6.12 and DLMF 22.2 for more information.
The Jacobian elliptic function \({\rm sn}(u,m).\)
The Jacobian elliptic function \({\rm cn}(u,m).\)
The Jacobian elliptic function \({\rm dn}(u,m).\)
The Jacobian elliptic function \({\rm ns}(u,m) = 1/{\rm sn}(u,m).\)
The Jacobian elliptic function \({\rm sc}(u,m) = {\rm sn}(u,m)/{\rm cn}(u,m).\)
The Jacobian elliptic function \({\rm sd}(u,m) = {\rm sn}(u,m)/{\rm dn}(u,m).\)
The Jacobian elliptic function \({\rm nc}(u,m) = 1/{\rm cn}(u,m).\)
The Jacobian elliptic function \({\rm cs}(u,m) = {\rm cn}(u,m)/{\rm sn}(u,m).\)
The Jacobian elliptic function \({\rm cd}(u,m) = {\rm cn}(u,m)/{\rm dn}(u,m).\)
The Jacobian elliptic function \({\rm nd}(u,m) = 1/{\rm dn}(u,m).\)
The Jacobian elliptic function \({\rm ds}(u,m) = {\rm dn}(u,m)/{\rm sn}(u,m).\)
The Jacobian elliptic function \({\rm dc}(u,m) = {\rm dn}(u,m)/{\rm cn}(u,m).\)
The inverse of the Jacobian elliptic function \({\rm sn}(u,m).\) For \(-1\le u \le 1,\) it can also be written (DLMF 22.15.E12):
The inverse of the Jacobian elliptic function \({\rm cn}(u,m).\) For \(-1\le u \le 1,\) it can also be written (DLMF 22.15.E13):
The inverse of the Jacobian elliptic function \({\rm dn}(u,m).\) For \(\sqrt{1-m}\le u \le 1,\) it can also be written (DLMF 22.15.E14):
The inverse of the Jacobian elliptic function \({\rm ns}(u,m).\) For \(1 \le u,\) it can also be written (DLMF 22.15.E121):
The inverse of the Jacobian elliptic function \({\rm sc}(u,m).\) For all \(u\) it can also be written (DLMF 22.15.E20):
The inverse of the Jacobian elliptic function \({\rm sd}(u,m).\) For \(-1/\sqrt{1-m}\le u \le 1/\sqrt{1-m},\) it can also be written (DLMF 22.15.E16):
The inverse of the Jacobian elliptic function \({\rm nc}(u,m).\) For \(1\le u,\) it can also be written (DLMF 22.15.E19):
The inverse of the Jacobian elliptic function \({\rm cs}(u,m).\) For all \(u\) it can also be written (DLMF 22.15.E23):
The inverse of the Jacobian elliptic function \({\rm cd}(u,m).\) For \(-1\le u \le 1,\) it can also be written (DLMF 22.15.E15):
The inverse of the Jacobian elliptic function \({\rm nd}(u,m).\) For \(1\le u \le 1/\sqrt{1-m},\) it can also be written (DLMF 22.15.E17):
The inverse of the Jacobian elliptic function \({\rm ds}(u,m).\) For \(\sqrt{1-m}\le u,\) it can also be written (DLMF 22.15.E22):
The inverse of the Jacobian elliptic function \({\rm dc}(u,m).\) For \(1 \le u,\) it can also be written (DLMF 22.15.E18):
The incomplete elliptic integral of the first kind, defined as
See also elliptic_e and elliptic_kc.
The incomplete elliptic integral of the second kind, defined as
See also elliptic_f and elliptic_ec.
The incomplete elliptic integral of the second kind, defined as
where \(\tau = {\rm sn}(u,m) .\)
This is related to elliptic_e
by
See also elliptic_e.
The incomplete elliptic integral of the third kind, defined as
The complete elliptic integral of the first kind, defined as
For certain values of \(m\), the value of the integral is known in
terms of \(Gamma\) functions. Use makegamma
to evaluate them.
The complete elliptic integral of the second kind, defined as
For certain values of \(m\), the value of the integral is known in
terms of \(Gamma\) functions. Use makegamma
to evaluate them.
Carlson’s RC integral is defined by
This integral is related to many elementary functions in the following way:
Also, we have the relationship
Some special values:
Carlson’s RD integral is defined by
We also have the special values
It is also related to the complete elliptic E function as follows
Carlson’s RF integral is defined by
We also have the special values
It is also related to the complete elliptic E function as follows
Carlson’s RJ integral is defined by