09/08/2009, 07:51 AM
(09/08/2009, 05:57 AM)Tetratophile Wrote: does the entire tetration cheat by using different branches of logarithm? like, for the sake of argument, tet(0)=1, but tet(-1)= \( 2k \pi i \) for some integer k
No, haha, it doesnt cheat. An entire function takes on every value infinitely often. Particularly the value 0, say sexp0(z_0)=0. Then you can just shift the function letting sexp(z)=sexp0(z0+1+z), then sexp(-1)=0. And by the equation sexp(z+1)=exp(sexp(z)) you get real values on all integers > -2.
Quote:why don't we adopt the entire tetration?
Actually Kneser constructed from that entire tetration a real analytic tetration (for details see my post with Kneser in the title).