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Old 2007-11-06, 23:48   #1
ShiningArcanine
 
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Default 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:
  • Why was I never taught about tetration in elementary school?
  • Are there rules for the differentiation and integration of tetration?
  • If there are 4 levels of arithmetic, is there a 5th level and if there is 5th level, what is it and are there infinitely many levels of arithmetic?

Last fiddled with by ShiningArcanine on 2007-11-06 at 23:50 Reason: Changed title
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Old 2007-11-07, 00:24   #2
Orgasmic Troll
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Quote:
Originally Posted by ShiningArcanine View Post
Why was I never taught about tetration in elementary school?
Because we usually like to teach things that we have convinced ourselves are useful. I don't know anyone using tetration on a day to day basis.

As far as differentiation, well, x^{u(x)} = e^{\log (x^{u(x)})} = e^{u(x) \log (x)} and \frac{d}{dx} e^{u(x) \log (x)} = (e^{u(x) \log (x)})(u'(x) \log (x) + \frac{u(x)}{x}) (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 2007-11-07 at 00:32
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Old 2007-11-07, 00:37   #3
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Okay, Mathematica gives numerical results, so there's some way to get approximate values, but I don't know much about numerical integration
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Old 2007-11-07, 00:45   #4
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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 non-elementary functions might be wrong.

Last fiddled with by ShiningArcanine on 2007-11-07 at 01:00
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