08/04/2014, 11:49 PM
(This post was last modified: 08/05/2014, 01:41 AM by sheldonison.)

(08/03/2014, 11:38 PM)tommy1729 Wrote: grows like .

Much much slower then exp or even its half-iterate.

And that for a function that is similarly defined. (second derivative = Original => second difference = Original )

Thats what I meant by counterintuitive.

I noticed the resemblance of the plots with the fake half-iterate exp too.

For the zero's Im convinced they are all real.

I made some numerical approximations, and for big enough values of x, using Jay's Taylor series, I get that , which match the equations in Gottfried's link. If ln(x)=1E100, then ln(f(x))~=7.213E199, ln(f(f(x)))~=3.753E399

So then . And yeah, that's growing much slower than

, if x is big enough.

This is despite many similarities in the complex plane graphs of f(x) and f(f(x)) as compared with the entire asymptotic to .

- Sheldon