Lim exp^[a](x) - exp( exp^[a-1](x) - exp^[a-1](x)^{-1} ) = 1. tommy1729 Ultimate Fellow Posts: 1,493 Threads: 356 Joined: Feb 2009 11/19/2015, 01:14 PM (This post was last modified: 11/19/2015, 01:26 PM by tommy1729.) Im recycling Some old Ideas I posted on sci.math. This one seems intresting. Let x be a positive real. Let a-1 >= 1. ^[*] is composition. Consider t(a) = $Lim \exp^{a}(x) - exp( exp^{a-1}(x) - exp^{a-1}(x)^{-1} ) = 1.$ Where the limit is for x going to + oo. For integer a , this is true and it can be easily proved by induction. In fact induction proves t(a+1) = t(a). Hence we get a periodic function for a-1 e [1,oo[. If we take f(a) = 1 for all real a ( a - 1 >= 1 ), What do we get ? A uniqueness condition together with D_x exp^[a-1](x) , (D_x)^2 exp^[a-1](x) > 0 for all real x ? What about $I(t) = lim \int_2^{n = oo} (t(a) - 1) / n da$ ? Is I(t) bounded from above ? From below ? Many questions from such a simple idea. Not even sure if we get analytic tetration. Regards Tommy1729 « Next Oldest | Next Newest »