09/06/2016, 03:47 PM
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 ...
---
Regards
Tommy1729
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 ...
---
Regards
Tommy1729