tetration

Half-iterate of exp(x).

**  The 2sinh method **

We consider 2 * sinh(x).

We take the half-iterate of it by using taylor series expanded at 0.

Thus f(x) is a taylor series , f(f(x)) = 2*sinh(x).

Clearly f(x) is then a good asymptote for the half iterate of exp(x) for large x.

Let g(g(x)) = exp(x)

Then g(x) =

Lim k-> oo   log log log log ...( k times ) (f( e^e^e^...( k times ) ^ x))

Clearly this can be generalized to tetration.

For a real 'r' to compute exp^[r] ( ^[] is iteration ) replace f(x) with (2*sinh(x))^[r] in the above formula.

Now you have ( for x and r being real ) exp^[r](x) aka tetration. It turns out this function is continuous and infinitely differentiable on the real axis (C^oo for fixed r and variable x). And it is probably analytic.

By analogue for a realvalued b we can compute (b^(x))^[r] and find the same properties as long as b > exp(1/2). 

All of this has been formally proven for b > exp(2) but Im working on the smaller bases too.

( One of the main reasons is that for b > exp(2) , 2*sinh(2x) has only one fixpoint (=0) whereas for exp(1/2) < b < exp(2) you have 3 fixpoints : zero and two nonreal complex conjugates )

Can this method be improved ?

YES !

    *** The exp(x) - exp(-3/5 x) method ***

Using f(x) = exp(x) - exp(-3/5 x) instead of 2sinh for the 2sinh method.

You might wonder why.

well the higher derivatives go into agreement with exp(x) for re(x) large unlike with 2sinh(x).

We also get the fixpoint at 0 and the function looks to go to exp in a more smooth way.

All the derivatives are positive at the origin and for x > 0.

This resembles exp(x) better thus in a way.

The solution is probably analytic and certainly well-behaved.

It also satisfies the semi-group isomo property. In fact it satisfies a monoid isomo property.

More precisely we have exp[a+b](x) = exp[a](exp[b](x)) = exp[b](exp[a](x)) for all real a,b > 0.

That property is a uniqueness condition !!

Notice that 

exp(x) - exp( - 1/3 x)  does NOT work since it has a fixpoint at some negative x.

This method is thus a consensus and compromise between using the fixpoint at 0 for exp(x) - 1 and 2sinh(x).

exp(x) - 1 is for base exp(1/e) however and is parabolic, so this method is more 2sinh type.

Notice that although this method is considered an improvement it is probably exactly the same function as with the 2sinh method.  This follows from the consideration that the 2sinh method also probably satisfies the monoid isomo property which is afterall a uniqueness condition. But it has benefits numerically and in terms of calculus.

***

Strongly related is the section " fake function theory ".

Also check out the sublinks.

***

All the above is for tetration base e.  Other bases and ideas are being investigated.

Below is a link to my profile at the tetration forum.

At the moment I only accept the tetration forum as a good source for tetration although you can find some info and links on wiki , I do however not accept the wiki page.

http://math.eretrandre.org/tetrationforum/member.php?action=profile&uid=47

or

https://tetrationforum.org/member.php?action=profile&uid=47

 (also posted at https://tetrationforum.org/showthread.php?tid=1750  on 05/16/2023, 11:30 PM )

Thank you for your intrest.

tommy1729