02/27/2022, 10:17 PM
Revitalizing an old idea : estimated fake sexp'(x).
It's an old idea but I want to put some attention to it again.
Perhaps with our improved skills and understanding this might lead us somewhere.
As most members and frequent readers know so-called fake function theory was developed by myself and sheldon to
1) create a real entire function
2) that is an asymptotic of a given function for positive reals
3) that has all its taylor or maclauren coefficients positive or at least non-negative.
There are some extra conditions such as strictly rising for positive real x imput for the given function and such.
But basicly that is what fake function theory is about.
We call such created functions - or the attempts - fake functions.
We also considered the cases were the given function already satisfied 1 -- 3 and how our algorithms created " fake ones ".
Many variants occured , such as replacing sums with analogue integrals and stuff but that is not important here.
We got good results for fake exp^[1/2](x).
So far the intro to the fake function theory part.
We also discussed base change constants here.
And we discussed the " expontential factorial " function ; the analogue of tetration like 2^3^4^... , the analogue of the factorial 1*2*3*4*... .
We are also familiar with telescoping sums.
And we even discussed converging infinite sums (over natural index n ) of the n th iteration of a function.
We also know the derivative of exp^[n](x) = exp^[n](x) * exp^[n-1](x) * exp^[n-2](x)*...
And how close this is related to the derivative of sexp(x) ; d sexp(x)/dx = sexp(x) * d sexp(x-1)/dx.
Is this all related to an old idea of myself and partially others ??
Yes certainly.
Lets estimate the derivative of sexp(x) but without using fake function theory tools directly.
Impossible ?
Well far from , if you accept brute estimates.
So we want something faster than any fixed amount of exp iterations.
But slower than sexp(x^2) or sexp(x)^2 or so.
The exact speed of the function is something to investigate and discuss.
But a logical attempt is this :
This function is entire ...
F3(x) = (1 + exp(x)/2^3) (1 + 2^3 exp^[2](x)/ 2^3^4 ) (1 + 2^3^4 exp^[3](x)/ 2^3^4^5 ) ...
and it converges fast.
It is also faster than any exp(x)^[k](x) for fixed k.
And all the coefficients are positive.
Also notice the pseudotelescoping product.
This leads to the asymptotic conjecture :
Using big-O notation :
For 1 < x
integral from 0 to x F3(t) dt = F4(x) = O ( fake sexp(x + c) ).
For some integration constant , and some constant c.
How about that ?
***
Next idea
lim v to oo ;
ln^[v] F4(x + v) = ??
Which also looks very familiar.
Regards
tommy1729
Tom Marcel Raes
It's an old idea but I want to put some attention to it again.
Perhaps with our improved skills and understanding this might lead us somewhere.
As most members and frequent readers know so-called fake function theory was developed by myself and sheldon to
1) create a real entire function
2) that is an asymptotic of a given function for positive reals
3) that has all its taylor or maclauren coefficients positive or at least non-negative.
There are some extra conditions such as strictly rising for positive real x imput for the given function and such.
But basicly that is what fake function theory is about.
We call such created functions - or the attempts - fake functions.
We also considered the cases were the given function already satisfied 1 -- 3 and how our algorithms created " fake ones ".
Many variants occured , such as replacing sums with analogue integrals and stuff but that is not important here.
We got good results for fake exp^[1/2](x).
So far the intro to the fake function theory part.
We also discussed base change constants here.
And we discussed the " expontential factorial " function ; the analogue of tetration like 2^3^4^... , the analogue of the factorial 1*2*3*4*... .
We are also familiar with telescoping sums.
And we even discussed converging infinite sums (over natural index n ) of the n th iteration of a function.
We also know the derivative of exp^[n](x) = exp^[n](x) * exp^[n-1](x) * exp^[n-2](x)*...
And how close this is related to the derivative of sexp(x) ; d sexp(x)/dx = sexp(x) * d sexp(x-1)/dx.
Is this all related to an old idea of myself and partially others ??
Yes certainly.
Lets estimate the derivative of sexp(x) but without using fake function theory tools directly.
Impossible ?
Well far from , if you accept brute estimates.
So we want something faster than any fixed amount of exp iterations.
But slower than sexp(x^2) or sexp(x)^2 or so.
The exact speed of the function is something to investigate and discuss.
But a logical attempt is this :
This function is entire ...
F3(x) = (1 + exp(x)/2^3) (1 + 2^3 exp^[2](x)/ 2^3^4 ) (1 + 2^3^4 exp^[3](x)/ 2^3^4^5 ) ...
and it converges fast.
It is also faster than any exp(x)^[k](x) for fixed k.
And all the coefficients are positive.
Also notice the pseudotelescoping product.
This leads to the asymptotic conjecture :
Using big-O notation :
For 1 < x
integral from 0 to x F3(t) dt = F4(x) = O ( fake sexp(x + c) ).
For some integration constant , and some constant c.
How about that ?
***
Next idea
lim v to oo ;
ln^[v] F4(x + v) = ??
Which also looks very familiar.
Regards
tommy1729
Tom Marcel Raes
