What is pi to the pi, pi times? (Was Q: `Half' a logarithm?)
John Chandler
Jack van Rijswijck <jav...@bausch.nl> wrote:
>Consider the function f(n,x) which is defined as follows
>for integer values of n:
>
>f(0,x) = x
>f(n+1,x) = exp(f(n,x))
>
>In other words:
>
>f(1,x) = exp(x)
>f(2,x) = exp(exp(x))
>f(3,x) = exp(exp(exp(x)))
>etc
>
>f(-1,x) = log(x)
>f(-2,x) = log(log(x))
>f(-3,x) = log(log(log(x)))
>etc
>
>How could this be extended to non-integer values of n? For
>example, if g(x) = f(0.5,x), then g satisfies: g(g(x)) = exp(x).
>So g is the function that exponentiates x "one half time". I
>found out that lim (x -> -oo) g(x) = -0.69608... which means
[munch]
>Is anything known about these funtions?
>
>
>Jack van Rijswijck
>jav...@ib.com
A related question that (allegedly) _has_ been answered is
how to extend the "tower" function to non-integer values of n:
Let
f(1,x) = x
f(2,x) = x^x
f(3,x) = x^x^x (by which is meant x^(x^x))
etc.
This defines f(n,x) for positive integer values of n.
How can f be extended as smoothly as possible
to non-integer values of n?
(Smoothness must be mentioned, as otherwise, whatever f(n,x)
is developed, we could add C*sin(pi*n) to it without changing
the values of f for integer values of n.
For smoothness, we might require that f and all of its
partial derivatives with respect to n be monotonically increasing,
for n greater than or equal to one. Or something...)
In particular, what is f(pi,pi)?
I heard this problem about forty years ago in the form
"What is pi to the pi, pi times?"
Supposedly this has been solved,
and f(pi,pi) is between 500,000 and 600,000,
but I don't know where it was published.
I once sent this problem in to Richard K. Guy for the Unsolved Problems
column in the Math Monthly. It was rejected as having been published,
but Guy was unable to tell me where.
Where is it???!??
If we can find that, it shouldn't be too hard to answer
van Rijswijck's question.
--
John Chandler
j...@a.cs.okstate.edu