section e of routines in elliptic.i

functions in elliptic.i - e

ell_am


    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.  
Interpreted function, defined at i/elliptic.i   line 93

SEE ALSO: elliptic, ellip_e, ell_f

elliptic


    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)  
Keyword,  defined at i/elliptic.i   line 10

SEE ALSO: ell_am, ell_f, ell_e, dn_, ellip_k, ellip_e