Dan Asimov wrote:
P.S. In a faintly related vein, if we define i^z := exp(pi*z*i/2), then iterating this function on the starting value of z = i approaches a limit L of approx. 0.6528812343931018 + i 0.3675743023883531 (or so says my C program).
Hmmm, Mathematica seems to give a rather different answer.
You didn't tell what Mathematica gave. I hope it was approx 0.438283 + i 0.360592 which would then agree with the exact expression I give below.
(Can someone please recompute; my program iterated i^z on the starting value z = i forty times before the new value was within 10^(-9) of the last one, but maybe my computation was done in by roundoff error.)
In any case:
1. Is it possible that this limit L of towers of exponentiated i's can be identified as some familiar number?
Isn't it just 2 i/pi W(-i pi/2) where W denotes the principal branch of the Lambert W function? David