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Lim exp^[a](x) - exp( exp^[a-1](x) - exp^[a-1](x)^{-1} ) = 1.
#1
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) =



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 ?

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
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