another thing: the derivation of x^x gives x^x*(1+Log(x)) it has thus a critical value at x_crit = Exp[-1]
the second derivative is x^(-1 + x) + x^x (1 + Log(x))^2 which is about
1.8816 at x_crit so it is a minima.
thinking of x^x as a continous function, 0^0 is thus clearly 1 ;-)
(see the silly plot in the attachement)
cu
On Wednesday 01 June 2011 20:56:52 Tobia Conforto wrote:
> Hi guys,
>
> it’s not strictly about Chicken, but a quick note about 0^0 is in order.
>
> Dominic Pearson wrote:
> > Hello folks,
> > I am trying to compute the sum i = 0 to n where n = 1000 of n^n
>
> Panos Stergiotis wrote:
> > Total[Table[i^i, {i, 1, 1000}]] + 1 yields
> > [...]
> > The + 1 is because 0^0 is not defined in mathematica
>
> It’s more than that, 0^0 is not defined in Maths.
>
> Simply put:
>
> lim(0^x) for x→0 = 0
> lim(x^0) for x→0 = 1
>
> Therefore 0^0 has more than one solution -or- is undefined.
>
> Graphical explanation: if you plot the surface z = x^y, you will see that
> it approaches the origin both with value constantly 1 from one direction
> (along the x axis, where y = 0) and with value constantly 0 from the other
> (along the y axis, where x = 0). Therefore, at the very origin, both 0 and
> 1 are solutions! Moreover, all numbers between 0 and 1 seem to be
> solutions too, looking at the graph. (Other numbers would be too, if you
> plotted the other quadrants of the surface… some of which are imaginary
> :-)
>
> Cheers
>
> Tobia
xx.pdf
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