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Re: Inverting elliptic integrals
Posted:
Aug 9, 2004 5:19 AM
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In article <muqqag9mwbv2@legacy>,
David Powers <powers@computer.org> 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,
Emphi := phi
-1/6*m*phi^3
+1/5*(1/6*m-1/8*m^2)*phi^5
+1/7*(-1/45*m+1/12*m^2-1/16*m^3)*phi^7
+1/9*(1/630*m-1/40*m^2+1/16*m^3-5/128*m^4)*phi^9 ...
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:
phi := E(phi,m)
+1/6*m*E(phi,m)^3
+1/120*m*(13*m-4)*E(phi,m)^5
+1/5040*m*(493*m^2-284*m+16)*E(phi,m)^7
+1/362880*m*(37369*m^3-31224*m^2+4944*m-64)*E(phi,m)^9 ...
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.
Richard J. Mathar, http://www.strw.leidenuniv.nl/~mathar
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