as for the " conjecture entire 1 "
things get intresting knowing that the entire functions grow like exp or faster despite being slower on the real line.
For instance fake logs or fake sqrt.
examples
sinh(sqrt(x)). (= fake of exp(sqrt(x)) )
sum x^n/(2^n!)
or this fake log here : http://math.eretrandre.org/tetrationforu...hp?tid=821
f(z) = ln( (2sinh(z) - z) / z )
--- edit ---
What Im trying to say here is this :
Despite entire superfunctions like that of 2sinh , " conjecture entire 1 " is still not formally decided.
I mentioned the weierstrass product , but however it gives imho a slight illusion.
The weierstrass product gives the impression that most entire functions that grow slower than exp(exp(x)) behave very much like exp(g(x)) p(x) where g(x) and p(x) are very close to polynomials.
( i sometimes call that conjecture 0 but that term has also been used for other conjectures )
The point of these " false functions " is to show this is not true !!
sinh(sqrt(z)) Always grows slower than about abs(exp(sqrt(abs(z)))) for abs(z). This follows from absolute convergeance ideas.
ln( (2sinh(z) - z) / z ) Always grows slower than about abs(ln( (2sinh(abs(z)) - abs(z)) / abs(z) )) for abs(z). This follows from absolute convergeance ideas.
So there is a " false function " for every elementary function.
( I still find this counterintuitive when I see weierstrass , sometimes even a proof is not convincing ? Maybe someone can enlighten me beyond proofs/examples )
SO why is there perhaps no " false function " for exp^[1/2] ?
Despite the arguments above, we still lack a proof , disproof or example.
So lets continue :
How - very sketchy - should such a " candidate function " look like ?
These candidate functions need to be between polynomial and exponential.
More precisely they need to be polynomial << f(z) << exp(ln(a) x^b).
So for instance f(z) ( the candidate ) cannot grow like exp(sqrt(x)-1).
We know from the previous posts that taking the half-iterate of a function close to exp makes a nice argument, but not a good example or counterexample.
( e.g. 2sinh^[1/2] )
2 Possibilities remain :
1) A power tower. Something like ln(x)^ln(ln(x))^... But thats even hard to prove being analytic.
2) sum z^n/(2^n!)
As a first remark I note that its not immediately clear - without alot of work - if a Taylor series is faster or slower than a power tower.
Second remark : x^ln(x) grows to slow : the growth = 0.
( actually the " fake x^ln(x) " to be exact )
This has already been demonstrated by me before.
This leads us back to remark 1 actually how do 1) , 2) , x^ln(x) relate ?
For the power tower case , I refer to the Original thread : http://math.eretrandre.org/tetrationforu...hp?tid=799
I will update that thread with a new idea. ( actually an old , but new here )
-------------------------------------------------------------------------------------
I conjecture a false exp^[1/2](z) = f(z) by chosing a suitable positive real A for
f(z)=0.5 + z + ( sexp(slog(1!)+A) )^-1 z^2 + ( sexp(slog(2!)+A) )^-1 z^3 + ... + ( sexp(slog((n-1)!)+A) )^-1 z^n
where n goes to +oo.
-------------------------------------------------------------------------------------
I think that expressed way better how I think about the subject then the post before.
So I edited.
regards
tommy1729
" Reality is that what does not go away when you stop believing in it "
things get intresting knowing that the entire functions grow like exp or faster despite being slower on the real line.
For instance fake logs or fake sqrt.
examples
sinh(sqrt(x)). (= fake of exp(sqrt(x)) )
sum x^n/(2^n!)
or this fake log here : http://math.eretrandre.org/tetrationforu...hp?tid=821
f(z) = ln( (2sinh(z) - z) / z )
--- edit ---
What Im trying to say here is this :
Despite entire superfunctions like that of 2sinh , " conjecture entire 1 " is still not formally decided.
I mentioned the weierstrass product , but however it gives imho a slight illusion.
The weierstrass product gives the impression that most entire functions that grow slower than exp(exp(x)) behave very much like exp(g(x)) p(x) where g(x) and p(x) are very close to polynomials.
( i sometimes call that conjecture 0 but that term has also been used for other conjectures )
The point of these " false functions " is to show this is not true !!
sinh(sqrt(z)) Always grows slower than about abs(exp(sqrt(abs(z)))) for abs(z). This follows from absolute convergeance ideas.
ln( (2sinh(z) - z) / z ) Always grows slower than about abs(ln( (2sinh(abs(z)) - abs(z)) / abs(z) )) for abs(z). This follows from absolute convergeance ideas.
So there is a " false function " for every elementary function.
( I still find this counterintuitive when I see weierstrass , sometimes even a proof is not convincing ? Maybe someone can enlighten me beyond proofs/examples )
SO why is there perhaps no " false function " for exp^[1/2] ?
Despite the arguments above, we still lack a proof , disproof or example.
So lets continue :
How - very sketchy - should such a " candidate function " look like ?
These candidate functions need to be between polynomial and exponential.
More precisely they need to be polynomial << f(z) << exp(ln(a) x^b).
So for instance f(z) ( the candidate ) cannot grow like exp(sqrt(x)-1).
We know from the previous posts that taking the half-iterate of a function close to exp makes a nice argument, but not a good example or counterexample.
( e.g. 2sinh^[1/2] )
2 Possibilities remain :
1) A power tower. Something like ln(x)^ln(ln(x))^... But thats even hard to prove being analytic.
2) sum z^n/(2^n!)
As a first remark I note that its not immediately clear - without alot of work - if a Taylor series is faster or slower than a power tower.
Second remark : x^ln(x) grows to slow : the growth = 0.
( actually the " fake x^ln(x) " to be exact )
This has already been demonstrated by me before.
This leads us back to remark 1 actually how do 1) , 2) , x^ln(x) relate ?
For the power tower case , I refer to the Original thread : http://math.eretrandre.org/tetrationforu...hp?tid=799
I will update that thread with a new idea. ( actually an old , but new here )
-------------------------------------------------------------------------------------
I conjecture a false exp^[1/2](z) = f(z) by chosing a suitable positive real A for
f(z)=0.5 + z + ( sexp(slog(1!)+A) )^-1 z^2 + ( sexp(slog(2!)+A) )^-1 z^3 + ... + ( sexp(slog((n-1)!)+A) )^-1 z^n
where n goes to +oo.
-------------------------------------------------------------------------------------
I think that expressed way better how I think about the subject then the post before.
So I edited.
regards
tommy1729
" Reality is that what does not go away when you stop believing in it "
