If you admire the beauty of mathematics and appreciate the ubiquitous number e, you'll enjoy a look at the function
Two notational observations are in order. First, following the usual convention, exponentiation "associates from the right", meaning a**b**c = a**(b**c). Second, by "..." we mean the limit (if one exists) as the number of exponentiations increases without bound.
This function is defined over a limited domain, as shown in the plot to the right. The coordinates of the endpoints use e in a delightful variety of ways.
To understand the overall shape of the curve, note that
if y = x**x**x**... exists, then x**y = y,
so x = y**(1/y). That curve looks like this:
As you see, in the region y > e the curve doubles back on itself,
(becoming asymptotic to the line x = 1),
so it doesn't represent a single-valued function of x.
Thus, it is not surprising that we will end up truncating
it at y = e, the point of maximal x; but it's less obvious
that it requires truncation for small y (and small x) as
well. For a discussion of convergence, continue
here.
This function was called to my attention in 1964 by Phil Rust. Discussions with Peter Ralph led to the discovery of the convergence bounds reported here.
2006-01-04
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