Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Actual formulas for tetration and its derivative
#11
Oh, right. Thanks. All the magic is gone now... Sad

As for my information, I use logic and intuition;
apparently both are on vacation today.
Sorry about that.

Andrew Robbins
Reply
#12
andydude Wrote:Oh, right. Thanks. All the magic is gone now... Sad

That was not my intention, perhaps it helps you that the first formula still has the same magic to me, as the second had to you Wink
Reply
#13
Well, unfortunately, no magic at all. Formulas 1 and 2 can be easily obtained, taking into account the hyperoperational properties of tetration, e.g..:
e#(x-1) = ln(e#x)
as well as the definition of the product logarithm:
if x . e^x = z then: x = ProductLog[z].

However, it is interesting to remind that the product logarithm is a complex function, with two real branches. I shall come back to it.

Please see the attached short pdf comment. I am too lazy to insert it in this text.

GFR


Attached Files
.pdf   No magic.pdf (Size: 5.96 KB / Downloads: 345)
Reply
#14
Hm, Gianfranco seems still to be a bit uncomfortable with the tex formulas. So it shall be forgiven that it came way to late, when everything was already solved Wink Tongue
Reply
#15
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
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Semi-exp and the geometric derivative. A criterion. tommy1729 0 1,122 09/19/2017, 09:45 PM
Last Post: tommy1729
  How to find the first negative derivative ? tommy1729 0 1,259 02/13/2017, 01:30 PM
Last Post: tommy1729
  A calculus proposition about sum and derivative tommy1729 1 1,735 08/19/2016, 12:24 PM
Last Post: tommy1729
  Derivative of exp^[1/2] at the fixed point? sheldonison 10 9,136 01/01/2016, 03:58 PM
Last Post: sheldonison
  Derivative of E tetra x Forehead 7 8,655 12/25/2015, 03:59 AM
Last Post: andydude
  A derivative conjecture matrix 3 4,075 10/25/2013, 11:33 PM
Last Post: tommy1729
  Non-recursive coefficient formulas. Can the Riemann mapping be constructed? mike3 0 1,899 06/04/2011, 12:17 AM
Last Post: mike3



Users browsing this thread: 1 Guest(s)