Using exp(x) - exp((1-e)*((x^3/1000) +x)) tommy1729 Ultimate Fellow     Posts: 1,491 Threads: 355 Joined: Feb 2009 10/03/2021, 10:03 PM 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  lim ln ln ln ... g^[y](exp(exp(...exp(x))) or  lim 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)) are 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 ? regards tommy1729 tommy1729 Ultimate Fellow     Posts: 1,491 Threads: 355 Joined: Feb 2009 10/03/2021, 10:05 PM (This post was last modified: 10/03/2021, 10:13 PM by tommy1729.) Notice this function - exp(x) - exp((1-e)*((x^3/1000) +x)) -  has only 1 real fixpoint. see also the 2sinh method : https://math.eretrandre.org/tetrationfor...hp?tid=424 regards tommy1729 JmsNxn Long Time Fellow    Posts: 566 Threads: 94 Joined: Dec 2010 10/05/2021, 12:17 AM (This post was last modified: 10/05/2021, 12:19 AM by JmsNxn.) 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. « Next Oldest | Next Newest »