A (probably wrong!) definition for a real height η, with 0 ≤ η < 1
A (probably wrong!) definition for a real height h, with 0 ≤ h < 1
Since we have
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if we define as
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it is tempting to consider (but probably wrong!) because is an exponent instead of a height in
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and
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That would have been too easy! — Daniel Forgues 02:09, 24 May 2013 (UTC) — Daniel Forgues 23:48, 25 May 2013 (UTC)
I'd like to know how this fails... — Daniel Forgues 02:37, 24 May 2013 (UTC)
- There are uncountably many possible definitions. I prefer to restrict definitions (at least when dealing with positive reals) to, say, [1, e] and extend it to other numbers by taking logs or exponents. Any "sensible" definition will be monotone, continuous, and have f(1) = 0, f(e) = 1.
- I assume what you intend is that for numbers in this range you take the logarithm, and add 1 each time you exponentiate so that the height of 4 is 2.3266 since log log 4 = 0.3266 (up to rounding).
- This has been suggested before, of course, though I don't remember the name usually associated with it. It's a bit ugly when graphed, which isn't unexpected: it's essentially defined piecewise, so whenever you switch from one piece to the next (at 1, e, e^e, e^e^e, etc.) the rate of growth suddenly increases.
- Charles R Greathouse IV 03:53, 25 May 2013 (UTC)
- I was worried that it might not have a continuous derivative (or maybe second derivative...) and we would have this piecewise behavior... That's ugly and feels very artificial. It should be defined in such a way that we get a smooth function (like the exponential function). How can we do it? — Daniel Forgues 07:48, 25 May 2013 (UTC)
- I've read a few papers that discuss getting a smooth function for tetration. All are quite ugly but some do exist. I don't have my binder of references handy at the moment, though, as I'm traveling this holiday weekend.
- Charles R Greathouse IV 13:47, 25 May 2013 (UTC)
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- For example, we have and thus
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- and (with the next definition, we get a smaller result)
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- Compare
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- For example, we have and thus
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- and (with the next definition, we get a smaller result)
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- Compare
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- Now, to evaluate the tetration with real height
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- since
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- we get
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- thus
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- and
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A (better?) definition for a real height w, with 0 ≤ w < 1
- I guess one could use some "kludge" to try to create a smooth (or maybe only having a continuous derivative?) function. Maybe some weighted geometric mean of a power tower of height (with weight ) and a power tower of height (with weight ), where and , e.g.
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- equivalent to
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- This might yield a continuous first derivative (progressively increasing rate of growth), although maybe not a continuous second derivative (the resulting function might not be smooth). I will search what people came up with as a smooth function for tetration... — Daniel Forgues 23:48, 25 May 2013 (UTC)
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This could be ill-defined, as the solutions might not be unique! — Daniel Forgues 23:48, 25 May 2013 (UTC) No problem: with the log, we get a linear equation. — Daniel Forgues 04:30, 27 May 2013 (UTC)
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- For example, we have and thus
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- giving
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- and (with the previous definition, we get a larger result)
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- Compare
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- For example, we have and thus
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- giving
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- and (with the previous definition, we get a larger result)
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- Compare
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- Now, to evaluate the tetration with real height
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- since
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- we get
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- thus
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- and
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- A definition for should at least be strictly increasing, continuous, and have with preferably a continuous first derivative, although it should much preferably be smooth (i.e. all derivatives continuous) since that is the case for whose derivatives are (Clarify: Any hint about what should be the derivatives of ? Maybe that could lead to ..) [1]
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- One would hope that the smoothness requirement might lead to a unique definition for .
- Also, considering that for the generalization of the factorial, the recursive condition
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- lead to a provably unique solution, namely the Gamma function, would then the recursive condition for the tetration
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- lead to a provably unique solution for the generalization of tetration?
- — Daniel Forgues 03:07, 27 May 2013 (UTC)
Notes
- ↑ Needs clarification (Any hint about what should be the derivatives of ? Maybe that could lead to .).— Daniel Forgues 02:20, 27 May 2013 (UTC)