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