## 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 93SEE 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 240SEE 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 180SEE 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 408SEE 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 391SEE 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 341SEE 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 303SEE 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 10SEE ALSO: ell_am, ell_f, ell_e, dn_, ellip_k, ellip_e