exponentiationfunctionsinfinity
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?
The problem is a bit messy since $f(x)$ is a step function. Initially, we can remove the steps with the restriction $x=90y$, under which conditions $f(y)=\ln (10y)! - \ln (9y)! - \ln (y)!$. We can now use Stirling's approximation to get $f(y)=10y\ln (10y)-10y -9y\ln (9y) +9y -y\ln y + y + O(\ln (10y)) = 10y \ln 10 + 10y \ln y -9y \ln 9 - 9y \ln y - y \ln y + O(\ln (10y)) = y(10 \ln 10 - 9 \ln 9) + O(\ln (10y))\approx 3.250829733 y + O(\ln (10y))$.
On the other hand, $g(y)=(2.07766)\sqrt{10}\sqrt{y}+(0.103883)(90y)=9.34947y+O(\sqrt{y})$.
Based on this, for at least those $x$ that are multiples of $90$, we can say that $g(x)$ grows faster than $f(x)$ (eventually), although they are both close to linear growth.
My favorite paper about $x^x$ is The $x^x$ Spindle, which appeared in Mathematics Magazine back in 1996. The main idea is to visualize the fact that we can write it as $$x^x = e^{x (\ln(x)+2k\pi i)}.$$ Note that for each choice of $k$, we get a different branch of the logarithm. Given any real number $x$, most of these branches will be complex valued. Thus, we can plot a curve in 3D. For $k=0$ (i.e., the principal branch), it looks something like so:
If we use more branches simultaneously, we get something like so:
If you plot the standard graph of $y=x^x$ including the values of $(p/q)^{p/q}$ for $p$ negative and $q$ odd and positive. Thus the graph might look something like so.
From the complex perspective, the dots arise as spots where one of the spiral threads punctures the $x$-$z$ plane.
As far as integration goes, Mathematica (which I used to make the images), doesn't return a value for the antiderivative - I'd be shocked if it can be expressed in closed form. Nonetheless, the integral can be expressed as a series (as in UserX's answer) and there's the fabulous fact, known as the sophomore's dream that $$\int_0^1 x^x \, dx = -\sum_{n=1}^{\infty} (-n)^{-n},$$ which was mentioned in the comments by Lucian.
Also, the integral can certainly be evaluated numerically. If we define $$F_k(x) = \int_{-\infty}^x e^{x (\ln(x)+2k\pi i)} \, dx,$$ Then the graphs of the resulting functions look something like so:
Best Answer
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 $