On 8/12/09, rcs@xmission.com <rcs@xmission.com> wrote:
Presumably you've noticed that all your denominators are 2K^2?
Rich
Ouch --- I hadn't --- thanks! But I had noticed that h(q) = h(p), where p = (2-2q)/(2-q) --- from which in particular the maximum h occurs where q = p, that is q = 2-sqrt2 = 0.585786438 ... though we still don't know exactly what h = 1.287188506 ... is here. And now following on from your remark, I notice that denom(h) = 2 (denom(p) denom(q))^2 in all cases [but denominators are always easier than numerators ...]. WFL Latest data: h(-sqrt2) = h(+sqrt2) = 0; h^2(-4/3) = h^2(7/5) = 59/450, h^2(-1) = h^2(4/3) = 11/18, h^2(-2/3) = h^2(5/4) = 287/288, h^2(-1/2) = h^2(6/5) = 231/200, h^2(-1/3) = h^2(8/7) = 1139/882, h^2(0) = h^2(1) = 3/2, h^2(1/4) = h^2(6/7) = 2511/1568, h^2(1/3) = h^2(4/5) = 731/450, h^2(2/5) = h^2(3/4) = 1311/800, h^2(1/2) = h^2(2/3) = 119/72. WFL