So, you want to give up the fun of thinking about this puzzle . Here is the

PART 1. Eye-to-eye with exponentiation...

This is why this sophisticated method is inferior to the brutal direct solution by iteration. Sophistication leads to a messy solution. It's so ironic.

However, the complex-plane map of the locations of such spurious solutions for our problem looks very pleasing, like a X-mas tree.

There is a way to distinguish the correct from wrong solutions, based on the convergence criterion discussed below, but that's another story.

The numerical solution

For instance, in IDL the program might look like:
__________________________________________________________
	print,' T^n:    n            real     imaginary'
	i=complex(0,1) & z=i
	for j=1,200 do begin
	        z=i^z
	        print,'T^',j,' =',z
        end
end
__________________________________________________________

and its output like:

 T^n:    n            real     imaginary
T^       1 =(     0.207880,      0.00000)
T^       2 =(     0.947159,     0.320764)
T^       3 =(    0.0500922,     0.602117)
T^       4 =(     0.387166,    0.0305271)
T^       5 =(     0.782276,     0.544607)
T^       6 =(     0.142562,     0.400467)
T^       7 =(     0.519786,     0.118384)
T^       8 =(     0.568589,     0.605078)
T^       9 =(     0.242365,     0.301151)
T^      10 =(     0.578489,     0.231530)
T^      11 =(     0.427340,     0.548231)
T^      12 =(     0.330967,     0.262892)
T^      13 =(     0.574271,     0.328716)
T^      14 =(     0.369948,     0.468173)
T^      15 =(     0.400633,     0.263120)
T^    
(...)
T^     198 =(     0.438283,     0.360593)
T^     199 =(     0.438283,     0.360592)
(...)
T^     inf =(     0.438283,     0.360592)

Is the p=0.438283+0.360593i a combination of well known transcendental numbers? Most probably not, but how can we prove it?

Otherwise, return to the home page of Pawel Artymowicz.

Last revised Feb. 1997