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