12/09/2007, 03:18 PM
Ivars Wrote:Question:
Do there exist any other roots z and y for equations:
h(z^y) = e^(i*pi/2+-I*2pi*k) ? for all k, or just for some single special k?
So we have in fact 3 pairs of z and y for which solutions exist:
z=e, y=pi/2 , z^y= e^pi/2 h(e^pi/2)= +-i
z=i, y=-i, z^y = i^-i= e^pi/2 h (i^-i)=+-i
z=-i, y=i, z^y = -i^i = e^pi/2 h(i^-i) = +-i
it also follows from formula h(1/(i^i)) = 1/i = -i
and h(1/(-i^-i)) = 1/-i = i.
so this formula h(1/z^z) =1/z must be valid not only for integer and real x, but also for imaginary and maybe complex z.
I find this very intriguing. But still do not know how to extend it beyond first 2 branches of W.
By the way, Gotfrieds curve which crosses imaginary axis at +- i seems to have minumal REAL value at
z= 25*pi=78,5398163397448
and is equal to h(25*pi). More exactly,
h(25pi) = -0,170356724523878+- I* 0,44394520769755.
I could not prove it, but it seems that:
h(25pi)^1/h(25pi) = 25 pi
h(25pi)^(-1/h(25pi)) = 1/(25*pi) =0,012732395
I wonder if other smaller curves on Gottfrieds initial plot ( where Im(h(z) = 0) might have crossing points with imaginary axis and real minimums which have also "nice" z values. If so, the whole curve might be a reasonably nice function- or is it already well known?