I'm having trouble parsing the EllipticE infinite series formula. Could you clarify if the 432^N is in the numerator or denominator, and similarly for the other terms? I suspect I don't understand the grouping of terms in Mma. Rich ---- Quoting Bill Gosper <billgosper@gmail.com>: <clippage -- rcs>

This means that EllipticE[1-r^2]==(2 \[Pi] Sum[-1/(n!)^2*432^n (((-1+r)^8 r (1+r)^2)/(1+60 r+134 r^2+60 r^3+r^4)^3)^n (-r (15+68 r+90 r^2+68 r^3+15 r^4)+n (1-126 r-1041 r^2-1764 r^3-1041 r^4-126 r^5+r^6)) Pochhammer[1/12,n] Pochhammer[5/12,n],{n,0,\[Infinity]}])/(1+60 r+134 r^2+60 r^3+r^4)^(5/4)

is an impressive acceleration formula for E'(r^2).

You might be tempted to object that the k-fold speedup is cancelled by the general term being k times more complicated, but this is not the case. Once r is fixed, this is just a matrix product over n of

quadratic/quadratic linear ( ). 1 0

However, it does cost a factor of 2 if you are computing megadigits of E'(nonsquare rational).

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

Which is, in fact, (nonobviously) symmetrical in a and b. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun