On 24.01.2012 22:52, math-fun-request@mailman.xmission.com wrote: ...

As you can see, the speed of convergence is pretty spectacular; 12 terms in each series will give you error<10^(-37).

Were Gosper to Rational-Remez these it should become slightly better still. In MAPLE, you allegedly can get it to find rational Remez best approximation of degree 7/degree 5 by with(numapprox); wght := (x) -> 1; minimax(q1, -1/4 - 10^(-87)..1/4 + 15.4632*10^(-88), [7, 5], wght, 'maxerror' ); #notice I have used a slight random enlargement of the #interval -1/4, 1/4 to prevent MAPLE #from activating some bullshit... but then it just gives different kind of failure

but in fact, I have never managed to make MAPLE actually succeed after several tries, gave up. MAPLE has a lot of bugs.

side remark (for q1(x) = (1/GAMMA(x+1/2)-1/GAMMA(-x+1/2))/x) and Maple) concerning Maple: it can be made to work, but needs high working precision (and I lost overview what is the one in the current discussion), Digits=8*16 will give an error of ~ 0.4*1e-20 using the command on - 1e-87 ... 1/4 + tiny and now using symmetry in zero gives it. I guess one reason for troubles is degree = (7,5), while the function is one in x^2. If writing it as r1(x):= q1( sqrt(abs(x)) ), i.e. q1(x) = r1(x^2) then the same command but for r1 with degree = (7,5) gives an error of 0.1*1e-31. With degree = (7,6), thus even degree as denominator, error is ~ 0.5*1-e36.