+- Tetration Forum (https://tetrationforum.org)
+-- Forum: Tetration and Related Topics (https://tetrationforum.org/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://tetrationforum.org/forumdisplay.php?fid=3)
+--- Thread: Lim exp^[a](x) - exp( exp^[a-1](x) - exp^[a-1](x)^{-1} ) = 1. (/showthread.php?tid=1035)
Lim exp^[a](x) - exp( exp^[a-1](x) - exp^[a-1](x)^{-1} ) = 1. - tommy1729 - 11/19/2015
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