In article <muqqag9mwbv2@legacy>, David Powers <email@example.com> writes: >I need the inverse of the (incomplete) elliptic integral of the second >kind. It is well defined but I can't track down a solution or code >for it. >....
For a numerical representation of the inverse in terms of the angle phi, where E(phi,m)=int(theta=0..phi) sqrt(1-m*sin^2 theta) dtheta is the elliptic integral fo the second kind, one could expand E(phi,m) in a power series around phi=0,
The expansion coefficients in front of the order phi^(n+1) (n=2,4,6,...) are [sum over k=2,4,..,n of U(k,n)*F(k,m)/k!]/(n+1)! where U(k,n) = (-1)^[(k+n)/2]/2^(k-n)*[sum l=0,1,...k of (-1)^l (l-k/2)^n*binom(k,l)] and F(k,m) = -m^(n/2)*[(k-1)!!]^2/(k-1) , with (k-1)!! = 1*3*5*7*...*(k-1) .
Then invert this as outlined in chapt 3.6.25 of the book edited by M Abramowitz and I Stegun:
Another efficient ansatz is a higher order Newton method, since the derivatives of E(phi,m) with respect to phi are well known: d E(phi,m)/d phi = sqrt(1-m*sin^2 phi) This needs in addition a solid implementation of the original E(phi,m) itself.