Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Actual formulas for tetration and its derivative
jaydfox Wrote:I've come up with these forumulae as well, though in a different form. They follow from basic principles and are "ignorant" of the underlying solution (i.e., the function for the critical interval doesn't matter).

To start with, let's bear in mind the following identity:

From this starting point, let's compute the first derivative using the chain rule:

From here, we simply rearrange to get .

As you can see, this is an identity, so there's no need to compute numerically to verify. If for some reason a particular solution doesn't satisfy this condition, then we can be sure it either doesn't satisfy the iterative exponential property, or it some other problem such as not being at least twice differentiable.

I haven't tried with the W function, but I would assume that the process is similar.
Oops. I wrote a subscript b, as though the base were arbitrary, but then I went ahead and assumed base e when I was solving. Amateur mistake.

Let's try again:

As before, we simply rearrange to get .

Again, as before, we have a recurrence relation we can leverage:

Et cetera.
~ Jay Daniel Fox

Messages In This Thread
RE: Actual formulas for tetration and its derivative - by jaydfox - 08/31/2007, 11:29 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Semi-exp and the geometric derivative. A criterion. tommy1729 0 1,415 09/19/2017, 09:45 PM
Last Post: tommy1729
  How to find the first negative derivative ? tommy1729 0 1,532 02/13/2017, 01:30 PM
Last Post: tommy1729
  A calculus proposition about sum and derivative tommy1729 1 2,084 08/19/2016, 12:24 PM
Last Post: tommy1729
  Derivative of exp^[1/2] at the fixed point? sheldonison 10 10,943 01/01/2016, 03:58 PM
Last Post: sheldonison
  Derivative of E tetra x Forehead 7 9,733 12/25/2015, 03:59 AM
Last Post: andydude
  A derivative conjecture matrix 3 4,693 10/25/2013, 11:33 PM
Last Post: tommy1729
  Non-recursive coefficient formulas. Can the Riemann mapping be constructed? mike3 0 2,322 06/04/2011, 12:17 AM
Last Post: mike3

Users browsing this thread: 1 Guest(s)