04/06/2009, 08:28 AM
tommy1729 Wrote:what is the area of that region?
That's not a stupid question, its actually a good question, and I don't know the answer. But let's see if we can answer that using what is known about this region. Galidakis (1) and I (2) both call it the Shell-Thron region since these two authors have both investigated this region in great detail. Shell and Thron note that \( {}^{\infty}b \) converges where \( b=h^{1/h} \) and \( |\ln(h)| \le 1 \) (a result which they both attribute to Barrow). So if we want the outer path, then we change the less-than sign to an equals sign, and this should give us the answer. If \( |\ln(h)| = 1 \), then we can parameterize this as \( \ln(h) = e^{it} \) where \( i=\sqrt{-1} \). This means that \( h = e^{e^{it}} \), and putting this back in the relationship with b gives the parameterization
\(
f(t) = {\left(e \uparrow e^{it}\right)} \uparrow {\left(e \uparrow {-e^{it}}\right)}
\)
where \( a \uparrow b = a^b \).f(t) = {\left(e \uparrow e^{it}\right)} \uparrow {\left(e \uparrow {-e^{it}}\right)}
\)
There are some interesting points that can be expressed with this function
\(
\begin{tabular}{rrl}
f(0) = & 1.4446678610097661337 & = e^{1/e} \\
f(1.4488307492834293737) =
& 2.0477905274644031305
& + i\ 0.842045503530840715 \\
f(1.927907601568660839
= & 0.8152027425068848021
& + i\ 2.0166402199122300356 \\
f(2.316910654383280043
= & -0.380979728647791984
& + i\ 0.8997001955459000918 \\
f(\pi) = & 0.0659880358453125371 & = e^{-e}
\end{tabular}
\)
While we could use these points to integrate each section of the region, we could also use the parametric integration formula
\(
A = \int_{x=\alpha}^{x=\beta} y dx = \int_{t=a}^{t=b} y(t) x'(t) dt
\)
where \( x(t) = \text{Re}(f(t)) \) and \( y(t) = \text{Im}(f(t)) \). Using numerical integration we find thatA = \int_{x=\alpha}^{x=\beta} y dx = \int_{t=a}^{t=b} y(t) x'(t) dt
\)
\(
\begin{tabular}{rl}
A
& = \int_{t=\pi}^{t=0} \text{Im}(f(t)) \left(\text{Re}(f(t))\right)^{'} dt \\
& = \frac{i}{2} \int_{h=1/e}^{h=e} h^{(1/h - 2)} \left(h^{1/h} - e^{1/(e^{1/\ln(h)} \ln(h))}\right) (\ln(h)-1) dh \\
& \approx 4.02546664
\end{tabular}
\)
which is only the top half of the region, and since A = 4.02546664046975481171259768713, then\begin{tabular}{rl}
A
& = \int_{t=\pi}^{t=0} \text{Im}(f(t)) \left(\text{Re}(f(t))\right)^{'} dt \\
& = \frac{i}{2} \int_{h=1/e}^{h=e} h^{(1/h - 2)} \left(h^{1/h} - e^{1/(e^{1/\ln(h)} \ln(h))}\right) (\ln(h)-1) dh \\
& \approx 4.02546664
\end{tabular}
\)
2A = 8.05093328093950962342519537425 should be the area of the whole region.
Andrew Robbins
(1) I.N.Galidakis The Birth of the Infinite Tetration Fractal.
(2) A.Robbins and H.Trappmann Tetration Reference, page 37.