Question about infinite tetrate

[tommy1729]

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 \( a \uparrow b = a^b \).

hmmm i was thinking about contour integration to arrive at a closed form ....