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@menu
* Introduction to Elliptic Functions and Integrals::
* Definitions for Elliptic Functions::
* Definitions for Elliptic Integrals::
@end menu
@node Introduction to Elliptic Functions and Integrals, Definitions for Elliptic Functions, , top
@comment node-name, next, previous, up
@section 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, 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
@math{m} instead of the modulus @math{k} or the modular angle
@math{\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:
@ifinfo
@math{m = k^2} and @math{k = \sin(\alpha)}
@end ifinfo
@tex
$$m = k^2$$ and $$k = \sin\alpha $$
@end tex
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:
@example
(%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)
@end example
Some examples of elliptic integrals:
@example
(%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))
@end example
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.
@node Definitions for Elliptic Functions, Definitions for Elliptic Integrals, Introduction to Elliptic Functions and Integrals, top
@comment node-name, next, previous, up
@section Definitions for Elliptic Functions
@defun JACOBI_SN (u, m)
The Jacobian elliptic function @math{sn(u,m)}.
@end defun
@defun JACOBI_CN (u, m)
The Jacobian elliptic function @math{cn(u,m)}.
@end defun
@defun JACOBI_DN (u, m)
The Jacobian elliptic function @math{dn(u,m)}.
@end defun
@defun JACOBI_NS (u, m)
The Jacobian elliptic function @math{ns(u,m) = 1/sn(u,m)}.
@end defun
@defun JACOBI_SC (u, m)
The Jacobian elliptic function @math{sc(u,m) = sn(u,m)/cn(u,m)}.
@end defun
@defun JACOBI_SD (u, m)
The Jacobian elliptic function @math{sd(u,m) = sn(u,m)/dn(u,m)}.
@end defun
@defun JACOBI_NC (u, m)
The Jacobian elliptic function @math{nc(u,m) = 1/cn(u,m)}.
@end defun
@defun JACOBI_CS (u, m)
The Jacobian elliptic function @math{cs(u,m) = cn(u,m)/sn(u,m)}.
@end defun
@defun JACOBI_CD (u, m)
The Jacobian elliptic function @math{cd(u,m) = cn(u,m)/dn(u,m)}.
@end defun
@defun JACOBI_ND (u, m)
The Jacobian elliptic function @math{nc(u,m) = 1/cn(u,m)}.
@end defun
@defun JACOBI_DS (u, m)
The Jacobian elliptic function @math{ds(u,m) = dn(u,m)/sn(u,m)}.
@end defun
@defun JACOBI_DC (u, m)
The Jacobian elliptic function @math{dc(u,m) = dn(u,m)/cn(u,m)}.
@end defun
@defun INVERSE_JACOBI_SN (u, m)
The inverse of the Jacobian elliptic function @math{sn(u,m)}.
@end defun
@defun INVERSE_JACOBI_CN (u, m)
The inverse of the Jacobian elliptic function @math{cn(u,m)}.
@end defun
@defun INVERSE_JACOBI_DN (u, m)
The inverse of the Jacobian elliptic function @math{dn(u,m)}.
@end defun
@defun INVERSE_JACOBI_NS (u, m)
The inverse of the Jacobian elliptic function @math{ns(u,m)}.
@end defun
@defun INVERSE_JACOBI_SC (u, m)
The inverse of the Jacobian elliptic function @math{sc(u,m)}.
@end defun
@defun INVERSE_JACOBI_SD (u, m)
The inverse of the Jacobian elliptic function @math{sd(u,m)}.
@end defun
@defun INVERSE_JACOBI_NC (u, m)
The inverse of the Jacobian elliptic function @math{nc(u,m)}.
@end defun
@defun INVERSE_JACOBI_CS (u, m)
The inverse of the Jacobian elliptic function @math{cs(u,m)}.
@end defun
@defun INVERSE_JACOBI_CD (u, m)
The inverse of the Jacobian elliptic function @math{cd(u,m)}.
@end defun
@defun INVERSE_JACOBI_ND (u, m)
The inverse of the Jacobian elliptic function @math{nc(u,m)}.
@end defun
@defun INVERSE_JACOBI_DS (u, m)
The inverse of the Jacobian elliptic function @math{ds(u,m)}.
@end defun
@defun INVERSE_JACOBI_DC (u, m)
The inverse of the Jacobian elliptic function @math{dc(u,m)}.
@end defun
@node Definitions for Elliptic Integrals, , Definitions for Elliptic Functions, top
@comment node-name, next, previous, up
@section Definitions for Elliptic Integrals
@anchor{ELLIPTIC_F}
@defun ELLIPTIC_F (phi, m)
The incomplete elliptic integral of the first kind, defined as
@ifinfo
@math{integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi)}
@end ifinfo
@tex
$$\int_0^\phi {{d\theta}\over{\sqrt{1 - m\sin^2\theta}}}$$
@end tex
See also @ref{ELLIPTIC_E}, @ref{ELLIPTIC_KC}
@end defun
@anchor{ELLIPTIC_E}
@defun ELLIPTIC_E (phi, m)
The incomplete elliptic integral of the second kind, defined as
@ifinfo
@math{ELLIPTIC_E(u, m) = integrate(sqrt(1 - m*sin(x)^2), x, 0, phi)}
@end ifinfo
See also @ref{ELLIPTIC_E}, @ref{ELLIPTIC_EC}
@end defun
@anchor{ELLIPTIC_EU}
@defun ELLIPTIC_EU (u, m)
The incomplete elliptic integral of the second kind, defined as
@ifinfo
@math{integrate(dn(v,m)^2,v,0,u) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2), t, 0, tau)}
where @math{tau = sn(u,m)}
@end ifinfo
@tex
$$\int_0^u {\rm dn}(v, m) dv = \int_0^\tau \sqrt{{1-m t^2}\over{1-t^2}} dt$$
where $\tau = {\rm sn}(u, m)$
@end tex
This is related to @math{ELLIPTIC_E} by
@ifinfo
@math{ELLIPTIC_EU(u, m) = ELLIPTIC_E(asin(sn(u,m)),m)}
@end ifinfo
@tex
$$E(u,m) = E(\phi, m)$$
where $\phi = \sin^{-1} {\rm sn}(u, m),m)$
@end tex
See also @ref{ELLIPTIC_E}.
@end defun
@defun ELLIPTIC_PI (n, phi, m)
The incomplete elliptic integral of the third kind, defined as
@ifinfo
@math{integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, 0, phi)}
@end ifinfo
@tex
$$\int_0^\phi {{d\theta}\over{(1-n\sin^2 \theta)\sqrt{1 - m\sin^2\theta}}}$$
@end tex
Only the derivative with respect to @math{phi} is known by MAXIMA.
@end defun
@anchor{ELLIPTIC_KC}
@defun ELLIPTIC_KC (m)
The complete elliptic integral of the first kind, defined as
@ifinfo
@math{integrate(1/sqrt(1 - m*sin(x)^2), x, 0, %pi/2)}
@end ifinfo
@tex
$$\int_0^{{\pi}\over{2}} {{d\theta}\over{\sqrt{1 - m\sin^2\theta}}}$$
@end tex
For certain values of @math{m}, the value of the integral is known in
terms of @math{Gamma} functions. Use MAKEGAMMA to evaluate them.
@end defun
@anchor{ELLIPTIC_EC}
@defun ELLIPTIC_EC (m)
The complete elliptic integral of the second kind, defined as
@ifinfo
@math{integrate(sqrt(1 - m*sin(x)^2), x, 0, %pi/2)}
@end ifinfo
@tex
$$\int_0^{{\pi}\over{2}} \sqrt{1 - m\sin^2\theta} d\theta$$
@end tex
For certain values of @math{m}, the value of the integral is known in
terms of @math{Gamma} functions. Use MAKEGAMMA to evaluate them.
@end defun
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