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Using exp(x) - exp((1-e)*((x^3/1000) +x))
Many ideas about tetration are not even published.
We doubt about our methods or the conjectured properties etc.

But I think it is time I post one of those ideas I was holding back.

Even before I joined the tetration forum I had the idea of using asymptotics to exp(z).

I think this applies to all of us here.

This eventually led to posting my 2sinh method.

The 2sinh method uses 2*sinh(y*z) to approximate exp(y*z).

because the derivative of 2*sinh(y*z) at its fixpoint 0 is 2*y  this method is limited to bases > exp(1/2) whereas we want tetration for all bases larger than exp(1/e) ( the so-called eta constant ).

Also better approximations and faster methods would be nice - at a practical computational price we pay though , but in theory they should have these properties -.

All of these improvements should be in any claimed improvement of the 2sinh method.

Better approximations and faster methods seems to be equivalent.

So I came up with this :

g(x) = exp(x) - exp((1-e)*((x^3/1000) +x))

The taylor series starts : g(x) = e x + ...

So the base problem is solved and we use the fixpoint at 0.

I take x,y as positive reals here just as with the 2sinh method.

Then we use one of 

ln ln ln ... g^[y](exp(exp(...exp(x)))


ln^[n] ( g^[y+n](x) )

from which we can compute a tetration function 
and compute the taylor series.

And then we use analytic continuation to extend to the complex plane.

Notice that exp(x) - exp((1-e)*((x^3/1000) +x)) is a good asymptotic to exp(x) , but not everywhere.
Then again that applies to 2sinh(x) as well and it is always true for entire asymptotics.


Ok so this is nice , but what properties do we have ??

Is it analytic ??

Are there arguments not mentioned already for the 2sinh(x) method ?

What happens at infinite imaginary ?

I assume it can not be a periodic solution.


The primary fixpoints of 
exp(x) - exp((1-e)*((x^3/1000) +x))


0.387845.. +/- 1.63884.. i

which is closer to the primary fixpoints of exp than the fixpoints of 2sinh are.
This gave me the hope and courage to post it.

the division by 1000 is for stability ; iterations and taylor coefficients seem nicer and more modest.
No formal reasons for that division by 1000.

What do you think ?


Notice this function - exp(x) - exp((1-e)*((x^3/1000) +x)) -  has only 1 real fixpoint.

see also the 2sinh method :


My guess is that this will converge to C^\infty, but not in the complex plane. I chop that up to it being too different from the iterates of the exponential.

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