03/21/2014, 11:55 PM

I was dreaming about uniqueness for a half-iterate of exp(x).

And what other properties it might have.

Let f ( f (x) ) = exp(x) where f (x) and f ' (x) are continu and increasing for all real x.

Define L1 as the limit for x -> +oo :

L1 = lim L2(x) / (a(x)+b(x))

where a(x) = x + f (x) , b(x) = sqrt(f (x)^2 + x^2)

and L2(x) = integral sqrt ( 1 + f ' (x)^2) dx where the integral goes from 0 to x.

The conjectured uniqueness is the lowest possible value for L1.

Thus let L be the lowest possible value for L1.

Then L = L1 and L has a unique function f (x) associated with it.

And what other properties it might have.

Let f ( f (x) ) = exp(x) where f (x) and f ' (x) are continu and increasing for all real x.

Define L1 as the limit for x -> +oo :

L1 = lim L2(x) / (a(x)+b(x))

where a(x) = x + f (x) , b(x) = sqrt(f (x)^2 + x^2)

and L2(x) = integral sqrt ( 1 + f ' (x)^2) dx where the integral goes from 0 to x.

The conjectured uniqueness is the lowest possible value for L1.

Thus let L be the lowest possible value for L1.

Then L = L1 and L has a unique function f (x) associated with it.