Searching for an asymptotic to exp[0.5]

[sheldonison]

One obvious questions from the Taylor series result, that I can't answer, because I have no idea how fast these functions grow as x goes to infinity, relative to exponentiaton. What is the "growth" of a functions like these, which should grow slower than exponentiation, but faster than any polynomial?
\( \sum_{n = 1}^{\infty}\frac{x^n}{(2n)!} \)
\( \sum_{n = 1}^{\infty}\frac{x^n}{(4n)!} \)

Wow -- both of these functions grow exactly exponentially, where growth is defined as slog(f^n)/n.

The first function is
\( f = \sum_{n = 1}^{\infty}x^{n}{(2n)!} = \cosh(\sqrt{x})-1\approx \frac{1}{2}\exp(\sqrt{x}) \)

But If my math is correct then iterating f is the same as iterating \( \exp(x/4) \), which is ..... the same as iterating an exponential!!!

The second function is
\( f = \sum_{n = 1}^{\infty}\frac{x^{n}}{(4n)!} \approx \frac{1}{4}\exp(x^{0.25}) \)

Ok, the second function is a little more complicated, but if my math is correct, it is going to be the same as iterating \( f(x)=\exp(\frac{x}{16}) \), which .... drumroll ... is also exponential growth!

So, half exponential functions, especially the Taylor series of entire versions of half exponential functions need more study... Lets conjecture that I have a constructive definition of an entire half exponential Taylor series, for which I haven't given all of the details, but I have a pari-gp program. Then the Taylor series coefficients must eventually grow slower than all of these entire functions with exponential growth.... really interesting!!!
- Sheldon