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17.1 Introduction to Elliptic Functions and Integrals | ||

17.2 Definitions for Elliptic Functions | ||

17.3 Definitions for Elliptic Integrals |

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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, Chapter 16-17. As much as possible, we use the definitions and relationships given there.

In particular, all elliptic functions and integrals use the parameter
*m* instead of the modulus *k* or the modular angle
*\alpha*. This is one area where we differ from Abramowitz and
Stegun who use the modular angle for the elliptic functions. The
following relationships are true:

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.

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__Function:__**jacobi_sn***(*`u`,`m`)The Jacobian elliptic function

*sn(u,m)*.

__Function:__**jacobi_cn***(*`u`,`m`)The Jacobian elliptic function

*cn(u,m)*.

__Function:__**jacobi_dn***(*`u`,`m`)The Jacobian elliptic function

*dn(u,m)*.

__Function:__**jacobi_ns***(*`u`,`m`)The Jacobian elliptic function

*ns(u,m) = 1/sn(u,m)*.

__Function:__**jacobi_sc***(*`u`,`m`)The Jacobian elliptic function

*sc(u,m) = sn(u,m)/cn(u,m)*.

__Function:__**jacobi_sd***(*`u`,`m`)The Jacobian elliptic function

*sd(u,m) = sn(u,m)/dn(u,m)*.

__Function:__**jacobi_nc***(*`u`,`m`)The Jacobian elliptic function

*nc(u,m) = 1/cn(u,m)*.

__Function:__**jacobi_cs***(*`u`,`m`)The Jacobian elliptic function

*cs(u,m) = cn(u,m)/sn(u,m)*.

__Function:__**jacobi_cd***(*`u`,`m`)The Jacobian elliptic function

*cd(u,m) = cn(u,m)/dn(u,m)*.

__Function:__**jacobi_nd***(*`u`,`m`)The Jacobian elliptic function

*nc(u,m) = 1/cn(u,m)*.

__Function:__**jacobi_ds***(*`u`,`m`)The Jacobian elliptic function

*ds(u,m) = dn(u,m)/sn(u,m)*.

__Function:__**jacobi_dc***(*`u`,`m`)The Jacobian elliptic function

*dc(u,m) = dn(u,m)/cn(u,m)*.

__Function:__**inverse_jacobi_sn***(*`u`,`m`)The inverse of the Jacobian elliptic function

*sn(u,m)*.

__Function:__**inverse_jacobi_cn***(*`u`,`m`)The inverse of the Jacobian elliptic function

*cn(u,m)*.

__Function:__**inverse_jacobi_dn***(*`u`,`m`)The inverse of the Jacobian elliptic function

*dn(u,m)*.

__Function:__**inverse_jacobi_ns***(*`u`,`m`)The inverse of the Jacobian elliptic function

*ns(u,m)*.

__Function:__**inverse_jacobi_sc***(*`u`,`m`)The inverse of the Jacobian elliptic function

*sc(u,m)*.

__Function:__**inverse_jacobi_sd***(*`u`,`m`)The inverse of the Jacobian elliptic function

*sd(u,m)*.

__Function:__**inverse_jacobi_nc***(*`u`,`m`)The inverse of the Jacobian elliptic function

*nc(u,m)*.

__Function:__**inverse_jacobi_cs***(*`u`,`m`)The inverse of the Jacobian elliptic function

*cs(u,m)*.

__Function:__**inverse_jacobi_cd***(*`u`,`m`)The inverse of the Jacobian elliptic function

*cd(u,m)*.

__Function:__**inverse_jacobi_nd***(*`u`,`m`)The inverse of the Jacobian elliptic function

*nc(u,m)*.

__Function:__**inverse_jacobi_ds***(*`u`,`m`)The inverse of the Jacobian elliptic function

*ds(u,m)*.

__Function:__**inverse_jacobi_dc***(*`u`,`m`)The inverse of the Jacobian elliptic function

*dc(u,m)*.

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__Function:__**elliptic_f***(*`phi`,`m`)The incomplete elliptic integral of the first kind, defined as

*integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi)*See also elliptic_e and elliptic_kc.

__Function:__**elliptic_e***(*`phi`,`m`)The incomplete elliptic integral of the second kind, defined as

*elliptic_e(u, m) = integrate(sqrt(1 - m*sin(x)^2), x, 0, phi)*See also elliptic_e and elliptic_ec.

__Function:__**elliptic_eu***(*`u`,`m`)The incomplete elliptic integral of the second kind, defined as

*integrate(dn(v,m)^2,v,0,u) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2), t, 0, tau)*where

*tau = sn(u,m)*This is related to

*elliptic_e*by See also elliptic_e.

__Function:__**elliptic_pi***(*`n`,`phi`,`m`)The incomplete elliptic integral of the third kind, defined as

*integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, 0, phi)*Only the derivative with respect to

*phi*is known by Maxima.

__Function:__**elliptic_kc***(*`m`)The complete elliptic integral of the first kind, defined as

*integrate(1/sqrt(1 - m*sin(x)^2), x, 0, %pi/2)*For certain values of

*m*, the value of the integral is known in terms of*Gamma*functions. Use`makegamma`

to evaluate them.

__Function:__**elliptic_ec***(*`m`)The complete elliptic integral of the second kind, defined as

*integrate(sqrt(1 - m*sin(x)^2), x, 0, %pi/2)*For certain values of

*m*, the value of the integral is known in terms of*Gamma*functions. Use`makegamma`

to evaluate them.

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