David Cantrell to the rescue. On Mon, Sep 26, 2011 at 3:33 AM, Bill Gosper <billgosper@gmail.com> wrote:
The Wikipedia ellipse article borrows (with the Goldwynism "For computational purposes a much faster series where the denominators vanish at a rate ...") a series from http://www.iamned.com/math/ which offers "A Rapidly converging formula for the circumference of an elipse that gives log (1024/27((a+b)/(a-b))^8) digits per term : ", a formula satisfyingly symmetrical in a and b, but which unfortunately vanishes (instead of 4a) for b=0,
This is an unrelated, 0^0 problem.
and generally fails for b<a.
No, I was using in Mma the formula based Maple's notion of elliptic E. (The old modulus vs parameter confusion.) Does anybody know what this should be? The Wikipedia article actually
corrects an obvious asymmetry in the iamned.com source, but the Wikigrapher must not have tested it.
Apologies: The Wikipedia formula is correct! (Except possibly at b=0.)
--rwg Here is what I typed: 8*\[Pi]/Q^(5/4)* Sum[Pochhammer[1/12, n]*Pochhammer[5/12, n]*(v1 + v2*n)* r^n/n!^2, {n, 0, \[Infinity]}] /. r -> 432*(a^2 - b^2)^2*(a - b)^6*b*a/Q^3 /. Q -> b^4 + 60*a*b^3 + 134*a^2*b^2 + 60*a^3*b + a^4 /. v1 -> a*b*(15*b^4 + 68*a*b^3 + 90*a^2*b^2 + 68*a^3*b + 15*a^4) /. v2 -> -a^6 - b^6 + 126*(a*b^5 + b*a^5) + 1041*(a^2*b^4 + a^4*b^2) + 1764*a^3*b^3
A correct formula for a>=b is elliplen[a_, b_] := 4*a*EllipticE[Sqrt[1 - b^2/a^2]]
No, elliplen[a_, b_] := 4*a*EllipticE[1 - b^2/a^2] --rwg