Re: Inverting elliptic integrals

From: David W. Cantrell (DWCantrell_at_sigmaxi.org)
Date: 07/29/04 

Date: 29 Jul 2004 22:45:04 GMT

wes <wesh@acm.org> wrote:
> In article <muqqag9mwbv2@legacy>, powers@computer.org says...
> > I need the inverse of the (incomplete) elliptic integral of the second
> > kind.

It has two arguments: its amplitude and its modulus.
I'd guess that you wish to invert it with respect to amplitude.

BTW, in case anyone reading this knows some references, I'd like
information about inverses of complete or incomplete elliptic integrals
with respect to _modulus_.

> > It is well defined but I can't track down a solution or code for it.
[snip]
> Go to functions.wolfram.com and all will be revealed.

As best I can tell, the only thing revealed by doing so is that there is no
pertinent information there.

Here's a liitle experiment using Mathematica and the On-line Encyclopedia
of Integer Sequences, first with an elliptic integral of the first kind,
then with one of the second kind:

In[1]:= InverseSeries[Series[EllipticF[z, m], {z, 0, 12}]]

Out[1]= z - (m*z^3)/6 + (1/120)*(4*m + m^2)*z^5 + ((-16*m - 44*m^2 -
m^3)*z^7)/5040 + ((64*m + 912*m^2 + 408*m^3 + m^4)*z^9)/362880 + ((-256*m -
15808*m^2 - 30768*m^3 - 3688*m^4 - m^5)*z^11)/39916800 + O[z]^13

If we then search the OEIS for, say, 1 4 1 44 16 1 408 912 64, we find an
entry related, not surprisingly, to a Jacobi elliptic function:

<http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A060627>.

In[2]:= InverseSeries[Series[EllipticE[z, m], {z, 0, 12}]]

Out[2]= z + (m*z^3)/6 + (1/120)*(-4*m + 13*m^2)*z^5 + ((16*m - 284*m^2 +
493*m^3)*z^7)/5040 + ((-64*m + 4944*m^2 - 31224*m^3 +
37369*m^4)*z^9)/362880 + ((256*m - 81088*m^2 + 1406832*m^3 - 5165224*m^4 +
4732249*m^5)*z^11)/39916800 + O[z]^13

If we then similarly search the OEIS for, say, either 13 4 493 284 16 or
4 13 16 284 493, we find ... nothing. This would not seem to bode well for
finding information about the desired inverse.

David Cantrell