20071106, 23:48  #1 
Dec 2005
2^{2}×23 Posts 
Tetration questions
My computer science professor showed us the Ackermann Function in my introductory C class today and said that if anyone wanted a challenge, he could write a iterative version of the Ackermann Function.
I started working on the task during class and I computed the Ackermann function for various values of n and m to get a feel for how I could write an iterative version. I derived 3 general formulas for the cases m = 1, m = 2 and m = 3 during class and after class, I realized that I forgot about m = 0 and that it was in the recursive definition of the function. Not long after I got home, I was dumbfounded when I derived a formula for m = 4, as I had no notation to represent it. I did not want to invent another notation (and name) for what I now know to be tetration, so I did a wikipedia search (i.e. I cheated) and I found the wikipedia page on the Ackermann function, which has an iterative version, a general formula for all values of the Ackermann function for m > 2, all of the general formulas I derived, the identities I had found when deriving the general formulas, etcetera. I have a few questions; the first was if there were any tetration identities, but Google seems to have answered it: http://tetration.itgo.com/ident.html If there are any not listed there, I would be happy to hear about them. My remaining questions are as follows:
Last fiddled with by ShiningArcanine on 20071106 at 23:50 Reason: Changed title 
20071107, 00:24  #2  
Cranksta Rap Ayatollah
Jul 2003
641 Posts 
Quote:
As far as differentiation, well, and (Hopefully, you can take it from here) as far as integration, well, at that point you take your toys and go home. ETA: Okay, maybe Monte Carlo integration would work .... Last fiddled with by Orgasmic Troll on 20071107 at 00:32 

20071107, 00:37  #3 
Cranksta Rap Ayatollah
Jul 2003
641 Posts 
Okay, Mathematica gives numerical results, so there's some way to get approximate values, but I don't know much about numerical integration

20071107, 00:45  #4 
Dec 2005
2^{2}×23 Posts 
Thanks for the differentiation rule for tetration. It is very clever. If I had tried to work it out, I would have went back to the definition of differentiation and tried to derive a general rule.
Anyway, I have a few more questions. We have sigma notation for series, which can collapse into multiplication if all of the terms are the same and product notation for series which can collapse into exponentiation if all of the terms are the same. Has anyone ever thought of using something like a T notation for series that can collapse into tetration if all of the terms are the same? If so, has anyone tried to find what I guess would be called an infinite exponentiation that is equal to some constant value, such as one? Edit: I was just thinking, e^(x^2) is an example of the series exponentiation I mentioned earlier in this post. It is not an elementary function and therefore it cannot be integrated by ordinary means, so I suppose it would stand to reason that all tetrations (where exponentiation is done more than once) are not elementary functions and therefore cannot be integrated by ordinary means. Another Edit: Looking at the definition of an elementary function on Wikipedia, my previous idea that tetrations with more than one exponentiation are nonelementary functions might be wrong. Last fiddled with by ShiningArcanine on 20071107 at 01:00 
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