12/16/2007, 12:02 PM

Gottfried Wrote:The found parameter-formula for s, based on the independent parameter beta, allowed then to find for an arbitrary s>e^(1/e) the according u and t (and even branches) employing regula falsi - just a poor-mans Lambert-W-replacement....

Gottfried

And my task was to find all real s>e^(1/e) for whom h(s) would be purely imaginary, Re(h(s))=0 as for h(e^(pi/2)) = +- i - the crossing points of Gottfrieds graphs with imaginary axis. Also the real minima they have ( negative). Analytically.

I have not suceeded so far. There fore I started to look into formulas involving h(z) hoping that will give answer eliminating need to compute all the steps and branches.

The best and true I have found is h(1/(n^n))=1/n for all n>=1 .

which leads to sum (h(1/(n^n))^2 = (pi^2)/6 etc. for other powers of h of 1/(n^n) = 1, 1/4, 1/27, 1/256, 1/3125, 1/46656, 1/823543, 1/16777216, 1/387420489, 1/10^10.............

These are like infinite polynomial equations h has to satisfy.

Other formulas are conjectures still.

Ivars