Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Exact and Unique solution for base e^(1/e)
For the rest of this post, assume .

This is mainly a theoretical solution. It'll be slow to converge without using helper functions to speed up certain processes. I'll cover some of the helper functions I've come up with, but in a later post.

For an arbitrary large integer n, we can find the tetration, calling it z:

It will be much easier to understand the following if we factor out the e.

Let's find the first logarithm. Remember the formula for logarithms in bases other than e:

Now, assuming z = e(1-d), let's see what we know about its logarithm and its second iterated logarithm:

Note that as n grows arbitrarily large, we see that on the interval y in [n-2,n], x^^y is must be almost linear, because we have three points that are almost colinear:

If you can't tell at a glance that the interpolating function must be linear in the limit as epsilon goes to 0, let me know. However, I assume that at this point, I've made my case. The interpolating function is linear, and since we're using tetration to integer hyperpowers to find the endpoints of our interval, we can solve exactly. We then use iterative logarithms (integer iteration count, again, so we can solve exactly).

It's important that everybody can agree that this solution is both exact and unique. The uniqueness of this solution can then be used to solve other bases, with some hope that those solutions will be unique as well. In other words, other solutions might meet the iterated exponential property and be infinitely differentiable, but they would be off by some cyclic factor in the tetrational exponent.

Given that I only have a bachelor's degree in computer science, with no formal graduate math study, I can only assume this exact solution must be known.

However, I haven't found it in my cursory reading of what's available on the internet, so apparently this solution, assuming it's already known, is buried in the literature. Perhaps it's considered too technical for us mere mortals. Anyway, can someone please point me to where this has been independently derived? Thanks.

(There is, of course, a chance that my solution is not unique, which is to say, that I've overlooked some small detail. But I really don't see that I could have missed anything, unless my series expansion for the second logarithm is wrong. However, I've triple-checked it.)

For background reference to how I derived this solution, see my sci.math.reference posts here:

Messages In This Thread
Exact and Unique solution for base e^(1/e) - by jaydfox - 08/10/2007, 03:59 AM

Possibly Related Threads...
Thread Author Replies Views Last Post
  tommy's simple solution ln^[n](2sinh^[n+x](z)) tommy1729 1 2,561 01/17/2017, 07:21 AM
Last Post: sheldonison
  Further observations on fractional calc solution to tetration JmsNxn 13 14,145 06/05/2014, 08:54 PM
Last Post: tommy1729
  Seeking a solution to a problem Qoppa/ssqrtQoppa=2 0 2,097 01/13/2011, 01:15 AM
Last Post: Qoppa/ssqrtQoppa=2
  Alternate solution of tetration for "convergent" bases discovered mike3 12 18,399 09/15/2010, 02:18 AM
Last Post: mike3
  Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation Base-Acid Tetration 19 30,880 10/24/2009, 04:12 AM
Last Post: andydude
  tetration solution f(b) tommy1729 0 2,153 09/17/2009, 12:28 PM
Last Post: tommy1729
  A tiny base-dependent formula for tetration (change-of-base?) Gottfried 8 12,314 03/18/2009, 07:26 PM
Last Post: Gottfried
  Superlog with exact coefficients andydude 7 8,036 03/13/2009, 06:14 PM
Last Post: bo198214
  Unique Holomorphic Super Logarithm bo198214 3 5,889 11/24/2008, 06:23 AM
Last Post: Kouznetsov

Users browsing this thread: 1 Guest(s)