Talk:Iterated logarithm

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

$0<\underbrace {\log \cdots \log \log \log \log } _{\log ^{*}x}\,x\leq 1,$

if we define $1-h$ as

$1-h:=\underbrace {\log \cdots \log \log \log \log } _{\log ^{*}x}\,x,$

it is tempting to consider (but probably wrong!) because $1-h$ is an exponent instead of a height in

$x\,{\overset {?}{=}}\,\underbrace {{e}^{{.\,}^{{.\,}^{{.\,}^{{e}^{{e}^{{e}^{e^{(1-h)}}}}}}}}} _{\log ^{*}x{\rm {~e's}}},\quad 0\leq h<1,$

and

$\operatorname {slog} \,x:=-h+\log ^{*}x.$

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 $\log ^{*}4=2$ and thus

$1-h=\log \log 4=0.32663425997828\ldots$

and (with the next definition, we get a smaller result)

$\operatorname {slog} \,4:=-h+\log ^{*}4=1.3266342599783\ldots$

Compare

$e\uparrow \uparrow 1=2.718281828459\ldots ,\,e\uparrow \uparrow 2=15.154262241479\ldots$

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For example, we have $\log ^{*}\pi =2$ and thus

$1-h=\log \log \pi =0.13516870162053\ldots$

and (with the next definition, we get a smaller result)

$\operatorname {slog} \,\pi :=-h+\log ^{*}\pi =1.1351687016205\ldots$

Compare

$e\uparrow \uparrow 1=2.718281828459\ldots ,\,e\uparrow \uparrow 2=15.154262241479\ldots$

Now, to evaluate the tetration with real height

$y=e\uparrow \uparrow 1.1351687016205\ldots \ldots =e\uparrow \uparrow (\operatorname {slog} \,y),$

since

$\operatorname {slog} \,y:=-h+\log ^{*}y=1.1351687016205\ldots ,\quad 0\leq h<1,$

we get

$\log ^{*}y=2,\,1-h=0.1351687016205\ldots ,$

thus

$\log \log y=1-h=0.1351687016205\ldots ,$

and

$y=e^{e^{(1-h)}}=3.1415926535897\ldots =\pi .$

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 $-1+\log ^{*}x$ (with weight $w$) and a power tower of height $\log ^{*}x$ (with weight $1-w$), where $0\leq w<1$ and $\operatorname {slog} \,x:=-w+\log ^{*}x$, e.g.

$x\,{\overset {?}{=}}\,\left(\underbrace {{e}^{{.\,}^{{.\,}^{{.\,}^{{e}^{{e}^{{e}^{e}}}}}}}} _{-1+\log ^{*}x}\right)^{w}\left(\underbrace {{e}^{{.\,}^{{.\,}^{{.\,}^{{e}^{{e}^{{e}^{e}}}}}}}} _{\log ^{*}x}\right)^{1-w},\quad x>1,\quad 0\leq w<1,$

equivalent to

$\log x\,{\overset {?}{=}}\,w\left(\underbrace {{e}^{{.\,}^{{.\,}^{{.\,}^{{e}^{{e}^{{e}^{e}}}}}}}} _{-2+\log ^{*}x}\right)+(1-w)\left(\underbrace {{e}^{{.\,}^{{.\,}^{{.\,}^{{e}^{{e}^{{e}^{e}}}}}}}} _{-1+\log ^{*}x}\right),\quad x>e,\quad 0\leq w<1.$

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)

This could be ill-defined, as the solutions $w$ 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 $\log ^{*}4=2$ and thus

$\log 4=1.3862943611199\ldots =w\,(e\uparrow \uparrow 0)+(1-w)\,(e\uparrow \uparrow 1)=w+(1-w)\,e=w\,(1-e)+e$

giving

$w=0.77518567983326\ldots$

and (with the previous definition, we get a larger result)

$\operatorname {slog} \,4:=-w+\log ^{*}4=1.2248143201667\ldots$

Compare

$e\uparrow \uparrow 1=2.718281828459\ldots ,\,e\uparrow \uparrow 2=15.154262241479\ldots$

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For example, we have $\log ^{*}\pi =2$ and thus

$\log \pi =1.1447298858494\ldots =w\,(e\uparrow \uparrow 0)+(1-w)\,(e\uparrow \uparrow 1)=w+(1-w)\,e=w\,(1-e)+e$

giving

$w=0.91577057764779\ldots$

and (with the previous definition, we get a larger result)

$\operatorname {slog} \,\pi :=-w+\log ^{*}\pi =1.0842294223522\ldots$

Compare

$e\uparrow \uparrow 1=2.718281828459\ldots ,\,e\uparrow \uparrow 2=15.154262241479\ldots$

Now, to evaluate the tetration with real height

$y=e\uparrow \uparrow 1.0842294223522\ldots =e\uparrow \uparrow (\operatorname {slog} \,y),$

since

$\operatorname {slog} \,y:=-w+\log ^{*}y=1.0842294223522\ldots ,\quad 0\leq w<1,$

we get

$\log ^{*}y=2,\,w=0.91577057764779\ldots ,$

thus

$\log y=w\,(e\uparrow \uparrow 0)+(1-w)\,(e\uparrow \uparrow 1)=w\,(1-e)+e=1.1447298858494\ldots$

and

$y=e^{1.1447298858494\ldots }=3.1415926535898\ldots =\pi .$

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A definition for $\operatorname {slog} \,x$ should at least be strictly increasing, continuous, and have $\operatorname {slog} \,1=0,\,\operatorname {slog} \,e=1,$ with preferably a continuous first derivative, although it should much preferably be smooth (i.e. all derivatives continuous) since that is the case for $\log x,\,x\in \mathbb {R} ^{+},$ whose derivatives are ^{(Clarify: Any hint about what should be the derivatives of $\operatorname {slog} \,x$? Maybe that could lead to $\operatorname {slog} \,x$..)} 

${\frac {{\rm {d}}^{k}\log x}{{\rm {d}}x^{k}}}={\frac {(-1)^{k-1}\,(k-1)!}{x^{k}}},\quad x\in \mathbb {R} ^{+},\,k\geq 1.$

One would hope that the smoothness requirement might lead to a unique definition for $\operatorname {slog} \,x$.

Also, considering that for the generalization of the factorial, the recursive condition

$f(x)=x\,f(x-1)$

lead to a provably unique solution, namely the Gamma function, would then the recursive condition for the tetration

$e\uparrow \uparrow x=e^{e\uparrow \uparrow (x-1)}$

lead to a provably unique solution for the generalization of tetration?

— Daniel Forgues 03:07, 27 May 2013 (UTC)