open problems survey

[JmsNxn]

Conjecture:

The standard extension of exponentiation is the only extension of exponentiation, such that a to the power of x is totally monotonic, For all real a between one and zero non inclusive, and for all real a.

OHHHHHH CATULLUS!!!

This is a good one.

I'd suggest an old thread by me on MathOverflow, but it's lost in all the overflow. To summarize the result:

Every totally monotonic function \(f\) can be described, using a unique measure \(\mu\), such:

\[
f(x) = \int_0^\infty e^{-xt}\,d\mu\\
\]

So... since \(f(x) = e^{-x}\) the exponential satisfies this formula for \(x>0\) when \(\mu(x) = \delta\) for the Dirac \(\delta\)-function, and we have a mole at this point. We've uniquely determined the exponential.

Essentially this means once you assume totally monotonic, you get a Laplace transform expression, then just use the Laplace transform to solve it. Since this process is reversible, it's unique.

Your conjecture is true.