GFR Wrote:- h: is used in the literature to indicate what I am calling "the infinite tower height" i.e.: h = (x->+oo)lim(b # x) = b # oo; we must observe that b # +oo can be real or complex, finite or infinite (variable with b), while b ^ +oo is always +oo, as far as I know, for b>0.
Hey Gianfranco, forgive me, but I have to correct:
\( b^\infty=0 \) for \( -1<b<1 \)
\( 1^\infty=1 \)
\( b^\infty=\infty \) for \( b>1 \).
And that we have two limit points in the case \( b=-1 \): \( (-1)^\infty=\pm 1 \); and two limit "points" for \( b<-1 \): \( b^\infty=\pm\infty \).
And there is really some similarity between exponentiation and tetration. The first similarity is when considering odd/even exponents. \( x^n \) and \( {^n}x \) is bijective exactly for odd exponents \( n \) (assuming that the domain is \( \mathbb{R} \) for \( x^n \) and \( \mathbb{R}_+ \) for \( {^nx} \), i.e. we map \( -\infty\mapsto 0 \), \( 0\mapsto 1 \)).
And this similarity continues to the limits, where we would map \( -\infty\mapsto 0 \), \( -1\mapsto e^{-e} \), \( 0\mapsto 1 \), \( 1\mapsto e^{1/e} \):
\( {^\infty b} \) has two limit points for \( 0<b< e^{-e} \)
\( {^\infty b} \) has one limit for \( e^{-e}\le b\le e^{1/e} \)
\( {^\infty b}=\infty \) for \( b>e^{1/e} \).
The difference in this analogy is that the behaviour directly at the borders is different. So \( {^\infty \left(e^{-e}\right)} \) has only one limit while \( (-1)^{\infty} \) has two limit points and the function \( {^\infty}x \) is continuous in \( x=e^{1/e} \), i.e. \( \lim_{x\uparrow e^{1/e}}{^\infty x}={^\infty(e^{1/e})} \) while the function \( x^{\infty} \) is not continuous in \( x=1 \), i.e. \( \lim_{x\uparrow 1} x^\infty =0\neq 1= 1^\infty \).
Quote:Then, attention, please! What I mean is that: I believe that function y = b # x, for b in the 0 .... Beta domain, oscillates, with oscillations between y/inf and y/sup, asymptotically decreasing towards h/inf and h/sup. These decreasing values are always and only verified for integer values of x (odd and even, respectively).
If we extend this analogy and asking ourselves why \( b^x \) has two limit points for \( b< -1 \) then the answer is not that \( b^x \) oscillates for \( b\le -1 \) but that it takes complex values there which just happen to be alternating real for natural \( x \). I pointed out a similar phenomenon for the function \( {^x}b \) somewhere already.