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Package elliptic (in elliptic.i) -
Index of documented functions or symbols:
DOCUMENT dn_(ell_am(u,m)) return the Jacobian elliptic function dn(u|m). The external variable ell_m must be set properly before calling dn_.
DOCUMENT ellip2_e(m)
returns the complete elliptic integral of the second kind E(M):
E(M) = integral[0 to pi/2]( dt * sqrt(1-M*sin(t)^2) )
accurate to 2e-8 for 0<=M<=1
DOCUMENT ellip2_k(m)
returns the complete elliptic integral of the first kind K(M):
K(M) = integral[0 to pi/2]( dt / sqrt(1-M*sin(t)^2) )
accurate to 2e-8 for 0<=M<1
DOCUMENT elliptic, ell_am, ell_f, ell_e, dn_, ellip_k, ellip_e
The elliptic integral of the first kind is:
u = integral[0 to phi]( dt / sqrt(1-m*sin(t)^2) )
The functions ell_f and ell_am compute this integral and its
inverse:
u = ell_f(phi, m)
phi = ell_am(u, m)
The Jacobian elliptic functions can be computed from the
"amplitude" ell_am by means of:
sn(u|m) = sin(ell_am(u,m))
cn(u|m) = cos(ell_am(u,m))
dn(u|m) = dn_(ell_am(u,m)) = sqrt(1-m*sn(u|m)^2)
The other nine functions are sc=sn/cn, cs=cn/sn, nd=1/dn,
cd=cn/dn, dc=dn/cn, ns=1/sn, sd=sn/dn, nc=1/cn, and ds=dn/sn.
(The notation u|m does not means yorick's | operator; it is
the mathematical notation, not valid yorick code!)
The parameter M is given in three different notations:
as M, the "parameter",
as k, the "modulus", or
as alpha, the "modular angle",
which are related by: M = k^2 = sin(alpha)^2. The yorick elliptic
functions in terms of M may need to be written
ell_am(u,k^2) or ell_am(u,sin(alpha)^2)
in order to agree with the definitions in other references.
Sections 17.2.17-19 of Abramowitz and Stegun explains these notations,
and chapters 16 and 17 present a compact overview of the subject of
elliptic functions in general.
The parameter M must be a scalar; U may be an array. The
exceptions are the complete elliptic integrals ellip_k and
ellip_e which accept an array of M values.
The ell_am function uses the external variable ell_m if M is
omitted, otherwise stores M in ell_m. Hence, you may set ell_m,
then simply call ell_am(u) if you have a series of calls with
the same value of M; this also allows the dn_ function to work
without a second specification of M.
The elliptic integral of the second kind is:
u = integral[0 to phi]( dt * sqrt(1-m*sin(t)^2) )
The function ell_e computes this integral:
u = ell_e(phi, m)
The special values ell_f(pi/2,m) and ell_e(pi/2,m) are the complete
elliptic integrals of the first and second kinds; separate functions
ellip_k and ellip_e are provided to compute them.
Note that the function ellip_k is infinite for M=1 and for large
negative M. The "natural" range for M is 0<=M<=1; all other real
values can be "reduced" to this range by various transformations;
the logarithmic singularity of ellip_k is actually very mild, and
other functions such as ell_am are perfectly well-defined there.
Here are the sum formulas for elliptic functions:
sn(u+v) = ( sn(u)*cn(v)*dn(v) + sn(v)*cn(u)*dn(u) ) /
( 1 - m*sn(u)^2*sn(v)^2 )
cn(u+v) = ( cn(u)*cn(v) - sn(u)*dn(u)*sn(v)*dn(v) ) /
( 1 - m*sn(u)^2*sn(v)^2 )
dn(u+v) = ( dn(u)*dn(v) - m*sn(u)*cn(u)*sn(v)*cn(v) ) /
( 1 - m*sn(u)^2*sn(v)^2 )
And the formulas for pure imaginary values:
sn(1i*u,m) = 1i * sc(u,1-m)
cn(1i*u,m) = nc(u,1-m)
dn(1i*u,m) = dc(u,1-m)
DOCUMENT ellip_e(m)
returns the complete elliptic integral of the second kind E(M):
E(M) = integral[0 to pi/2]( dt * sqrt(1-M*sin(t)^2) )
See help,elliptic for more.
DOCUMENT ellip_k(m)
returns the complete elliptic integral of the first kind K(M):
K(M) = integral[0 to pi/2]( dt / sqrt(1-M*sin(t)^2) )
See help,elliptic for more.
DOCUMENT ell_am(u)
or ell_am(u,m)
returns the "amplitude" (an angle in radians) for the Jacobi
elliptic functions at U, with parameter M. That is,
phi = ell_am(u,m)
means that
u = integral[0 to phi]( dt / sqrt(1-m*sin(t)^2) )
Thus ell_am is the inverse of the incomplete elliptic function
of the first kind ell_f. See help,elliptic for more.
SEE ALSO: elliptic