Home Manual Packages Global Index Keywords Quick Reference | # 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
ell_e |
                 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
SEE ALSO: | elliptic, ell_f
ell_f |
                 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
SEE ALSO: | elliptic, ell_e
ellip2_e |
                 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
SEE ALSO: | elliptic, ellip_k, ell_e
ellip2_k |
                 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
SEE ALSO: | elliptic, ellip_e, ell_f
ellip_e |
                 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
SEE ALSO: | elliptic, ellip_k, ell_e
ellip_k |
                 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
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