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I read on Wikipedia that growth rate of a function can sometimes be greater than exponential. Can you give me some examples of such functions (preferably continuous ones)?

Obviously $x^x$ grows faster than normal exponential, and $x^{x^x}$ even more so - does this concept have a name, and can an arbitrary/infinite amount of such "exponentiality" be expressed with a mathematical expression?

Any other interesting functions to be aware of?

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For the name of the concept: look for "powertower" in mathworld or wikipedia, and for attempts to the extension to continuous "heights" (the number of x in your question as the variable parameter) look for the term "tetration" in wikipedia (note, that there is not yet an accepted extension for general powertowers to interpolated "heights") –  Gottfried Helms Jan 25 '12 at 20:40
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3 Answers

up vote 3 down vote accepted

The Gamma function defined by $\Gamma(x) = \int_0^\infty e^{-t}t^{x-1}dt$ is a continuous function (and further, an analytic function) for which $\Gamma(n) = (n-1)!$ for all $n \in \mathbb{N}$. In particular, it grows faster than any exponent.

Asymptotically,

$\Gamma(z) \cong \sqrt{z} \cdot (\frac{z}{e})^z $

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You can compose the exponential function any number of times with itself, say $f_0(x)=x$ and $f_{n+1}(x)=\exp(f_n(x))$ for all $n\in\mathbf N$, to get ever faster growing functions. Every $f_n$ is an analytic function, so everywhere (in $\mathbf C$) indefinitely differentiable. For increasing arguments $x\in\mathbf R$, these functions still grow much slower than even $2\uparrow\uparrow m$ does as a function of $m\in\mathbf N$, because the latter composes $x\mapsto 2^x$ un unbounded number of times (namely $m$ times) with itself (and then applies to $1$), whereas each $f_n$ only has a fixed number $n$ compositions of $\exp$. Lacking a continuous version of function composition (composing a function $x$ times with itself), I'm not sure one can match the growth of $2\uparrow\uparrow m$ with an analytic function.

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$@$Marc: There is a standard result in complex analysis (in the neighborhood of Weierstrass Factorization and Mittag-Leffler) that given any closed discrete subset $\Omega \subset \mathbb{C}$ and any function $f: \Omega \rightarrow \mathbb{C}$, $f$ can be extended to an entire function. Taking $\Omega = \mathbb{N}$, we see that we can interpolate the up-arrow stuff by an analytic function. –  Pete L. Clark Jan 25 '12 at 19:41
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There is a standardized system of writing very large numbers, to be found here. The example from the page looks like this:

$$ 2 \uparrow\uparrow\uparrow 4 = \begin{matrix} \underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^2}}}}}}\\ \qquad\quad\ \ \ 65,536\mbox{ copies of }2 \end{matrix} \approx (10\uparrow)^{65,531}(6.0 \times 10^{19,728}) \approx (10\uparrow)^{65,533} 4.3 , $$ where $(10\uparrow)^n$ denotes a functional power of the function $f(n) = 10n$.

The $\uparrow$ is Knuth's up-arrow notation. With it $x^x$ translates to $x\uparrow\uparrow2$ and $x^{x^x}$ to $x\uparrow\uparrow3$.

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