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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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1  
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

4 Answers 4

up vote 4 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

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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Other than that, you can create any number of your own operattions that grow faster than exponential and produce continuous curves with infinite sums. You need not be thrown off by the fact that it's a discrete summation, any certified mathematician will confirm that an operator is an operator is an operator is an operator is an operator is an operator, and that specific infinite summation operation on "x" can be applied to any and all real numbers, including decimals and irrational numbers. The same exact principal is used for integrals, which as we all know are really the limits of discrete summations which produce plenty of continuous curves. Some of these summation operators have names, such as the Reimann Zeta function, Dirichlet L-function, the Hurwitz zeta function, any power series and anything you can think of that you test that grows faster than an exponential. In fact, here's a list of operators https://en.wikipedia.org/wiki/List_of_operators

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