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Searching for an asymptotic to exp[0.5]
Back to basics

In addition to post 17,18 notice that

D exp^[1/2](x) = D exp^[1/2] (exp(x) ) * exp(x) / exp^[1/2](exp(x)).

That follows from ln exp^[a] ( exp(x) ) = exp^[a](x) and the chain rule for derivatives.

By induction / recursion this gives a nice way ( product ) to compute the derivative.

This strenghtens the conclusions from post 17 , 18 and shows that

1 + o(1) <<_n 2.

( smaller after only a few iterations n )

We conclude by noting that the Taylor T

T = Sum_{K=4}^{oo} d_k x^k

With d_k = exp( - k^2 )

grows slower then exp^[1/2](x) , yet faster then any polynomial.

T has growth 0 , like exp( ln^2 (x) ) and similar.

Did we meet T yet ?? I believe T was a fake of Some elementairy like exp( ln^2 ) or such ...
This again leads to the desire of inverse fake or its related integral transforms ...

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Regards

Tommy1729


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I request complex plots of f(x) = fake exp^[1/3](x) , f(f(x)) and f(f(f(x))).

Like sheldon did for fake exp^[1/2](x) in one of the early posts in this thread.

It is very important !!

( potentially new results/conjectures based on those plots ! )

Regards

Tommy1729

Ps make sure to make backups of this website/content ? Bo ?
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