functions in elliptic.i - e

             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  

             ell_e(phi,m)  
 
     returns the incomplete elliptic integral of the second kind E(phi|M).  
     That is,  
        u = ell_e(phi,m)  
     means that  
        u = integral[0 to phi]( dt * sqrt(1-m*sin(t)^2) )  
     See help,elliptic for more.  
interpreted function, defined at i/elliptic.i   line 240  

             ell_f(phi,m)  
 
     returns the incomplete elliptic integral of the first kind F(phi|M).  
     That is,  
        u = ell_f(phi,m)  
     means that  
        u = integral[0 to phi]( dt / sqrt(1-m*sin(t)^2) )  
     See help,elliptic for more.  
interpreted function, defined at i/elliptic.i   line 180  

             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  
interpreted function, defined at i/elliptic.i   line 408  

             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  
interpreted function, defined at i/elliptic.i   line 391  

             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.  
interpreted function, defined at i/elliptic.i   line 341  

             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.  
interpreted function, defined at i/elliptic.i   line 303  

             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