12/17/2012, 04:34 PM
I was wondering about the following conjectured uniqueness criterion for real-analytic half iterates ( easily generalized to all iterates btw) :
(based upon my sinh method)
for all real x:
(D is the derivative operator)
D exp^[1/2](x) > 0
D^2 exp^[1/2](x) > 0
exp^[1/2](x) - ln(ln(2sinh^[1/2](exp(exp(x))))) > 0.
It seems intuitive , especially considering how the sinh method works.
However tetration is sometimes (or often) counterintuitive.
A stronger conjecture would be :
D exp^[1/2](x) > 0
D^2 exp^[1/2](x) > 0
ln(ln(2cosh^[1/2](exp(exp(x))))) > exp^[1/2](x) > ln(ln(2sinh^[1/2](exp(exp(x)))))
Although this requires some concept of cosh^[1/2].
( I once considered a method based upon cosh but made a mistake )
Im not even sure about the uniqueness of these uniqueness criterions.
For instance how this might be related to D^n exp^[1/2](x) > 0 for integer n with n>1 and/or curvature and/or length conditions.
Intuitively it seems like f(x) = exp^[1/2](x) computed by the sinh method gives the shortest path/length between f(a) and f(b) for sufficiently large real a and b but im not even sure if that is a selfconsistant statement for ANY exp^[1/2](x).
Also clearly shortest length and smallest curvature relate alot.
Recently I posted a new thread that involved the point
w := f ' (w) = 1
which might relate to uniqueness.
Clearly there is still work.
And I must admit its not all clear to me yet.
Tetration is calculus on drugs
regards
tommy1729
(based upon my sinh method)
for all real x:
(D is the derivative operator)
D exp^[1/2](x) > 0
D^2 exp^[1/2](x) > 0
exp^[1/2](x) - ln(ln(2sinh^[1/2](exp(exp(x))))) > 0.
It seems intuitive , especially considering how the sinh method works.
However tetration is sometimes (or often) counterintuitive.
A stronger conjecture would be :
D exp^[1/2](x) > 0
D^2 exp^[1/2](x) > 0
ln(ln(2cosh^[1/2](exp(exp(x))))) > exp^[1/2](x) > ln(ln(2sinh^[1/2](exp(exp(x)))))
Although this requires some concept of cosh^[1/2].
( I once considered a method based upon cosh but made a mistake )
Im not even sure about the uniqueness of these uniqueness criterions.
For instance how this might be related to D^n exp^[1/2](x) > 0 for integer n with n>1 and/or curvature and/or length conditions.
Intuitively it seems like f(x) = exp^[1/2](x) computed by the sinh method gives the shortest path/length between f(a) and f(b) for sufficiently large real a and b but im not even sure if that is a selfconsistant statement for ANY exp^[1/2](x).
Also clearly shortest length and smallest curvature relate alot.
Recently I posted a new thread that involved the point
w := f ' (w) = 1
which might relate to uniqueness.
Clearly there is still work.
And I must admit its not all clear to me yet.
Tetration is calculus on drugs
regards
tommy1729