PlanetMath: function $x^x$

(more info)

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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$} \end{pspicture}$

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