PlanetMath: function $x^x$

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

function

(Example)

Some properties of the real function $x \mapsto x^x$ :

The function is defined only for positive values of (and for negative integers which we neglect in the following items); cf. fraction power and power function.
can be composed of and $t = x\ln{x}$ .
$\displaystyle\lim_{x\to 0+}x^x = 1$ , as one sees by substituting $t = x\ln{x}$ into the Taylor expansion of .
is differentiable: Using the chain rule on $e^{x\ln{x}}$ , one obtains:
$\displaystyle \frac{d}{dx}(x^x) = x^x(1+\ln{x}).$
The function has absolute minimum value $\displaystyle e^{-\frac{1}{e}} \approx 0.6922$ and achieves all real values above this.
$\displaystyle\lim_{x\to\infty}\frac{e^x}{x^x} = \lim_{x\to\infty}e^{(1-\ln{x})x} = 0$ (cf. growth of exponential function)
has no elementary function as antiderivative.
$\displaystyle \int_0^1 x^x\,dx = \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^n} = 0.7834305107\ldots$
satisfies the differential equations
$\displaystyle yy'' - (y')^2 - y^2/x = 0$
and
$\displaystyle y^3 y''' - y^2 (y'')^2 + 2 y (y')^2 y'' - 3 y^2 y' y'' - (y')^4 + 2 y (y')^3 = 0.$

$\begin{pspicture}(-0.5,-0.5)(3.5,9) \psaxes[Dx=1,Dy=1]{->}(0,0)(-0.5,-0.5)(3.5,8... ....35,0.69) \rput(0.35,-0.36){$\frac{1}{e}$} \rput(3,5){$y = x^x$} \end{pspicture}$

"function