Question about tetration methods

[tommy1729]

Gottfried claimed that there were about fifteen approaches to tetration. Could someone list the more important techniques? Any additional information would be great.

I think you are confused.

I recently said that.

Which ones are more important , better , faster , more practical , analytic , equal etc can be debated !

Here are some for bases larger than 2 that immediately cross my mind :

1) the base change method.  and their matrix analogue ideas see https://math.eretrandre.org/tetrationfor...php?tid=26

One of the simplest I guess but many believe it is not analytic.

2) the slog computed by an infinite linear system ; Andrew's slog method ... Peter Walker seems to be the original inventor.

see https://math.eretrandre.org/tetrationfor....php?tid=3

and https://math.eretrandre.org/tetrationfor...p?tid=1081

3) The carleman matrix method. Gottfriend in particular seems to prefer this matrix method.

4) Kouznetsov's method. Based on contour integrals.

5) Kneser's solution. Based on the fixpoint of ln(z). Uses Riemann mapping and fourier series. Apparently proven analytic. 

6) the gaussian method. And similar ofcourse. Uses the erf function from gauss. Posted by me as an improved or alternative version of James Nixons beta method. see https://math.eretrandre.org/tetrationfor...p?tid=1339

7) Using 2 sinh(x) as an asymptotic. see : https://math.eretrandre.org/tetrationfor...hp?tid=424

and related : https://math.eretrandre.org/tetrationfor...hp?tid=953

8 ) Using continuum sums and continuum products.

9) using interpolation

10) this taylor type solution NOT REQUIRING A FIXPOINT that I posted :  https://math.eretrandre.org/tetrationfor...hp?tid=791

11) An infinite sum method I made somewhere.

12) Iteration approximations of exp and in the limit getting tetration ( assuming that works ! ).  Such as iterating the taylor polynomials of exp(z) by growing degree.

13) Using fixpoints at real +/- infinity or trying at least.

14) moving fixpoints of approximations to infinity such as : https://math.eretrandre.org/tetrationfor...p?tid=1400

15) https://math.eretrandre.org/tetrationfor...p?tid=1356
 
another approximation method.

16) a special kind of interpolation method :  https://math.eretrandre.org/tetrationfor...p?tid=1257

17) not analytic but worth mentioning maybe : https://math.eretrandre.org/tetrationfor...p?tid=1188

18 ) nonconstructive methods that seem to exist though.

19)  secondary fixpoints and merged fixpoints ideas.

and many more.

and apparantly there are others not even discussed at this forum.



I also like this but it is not really tetration : https://math.eretrandre.org/tetrationfor...p?tid=1013

also nice is this concept : https://math.eretrandre.org/tetrationfor...hp?tid=166


yes yes im aware all of these need explaining and links ... but that would be half the forum :/ 

and maybe im biased sorry.


regards

tommy1729