Searching for an asymptotic to exp[0.5]

[sheldonison]

Let f(z) be the entire fake half-iterate of exp(z).

Now we do not have a sexp and hence also no theta(z) function.

However at least at the positive real line there must other entire functions that satisfy the functional equation very well.
....
It seems u(z) makes a " wobble " just like we saw with the theta functions.

We could also look at \( u(z)=\log(f(f(z))-1\approx f(f(z))/\exp(z)-1 \)

Still on vacation; enjoyed reading your posts. I generated a quick graph of \( \log(|\frac{f(f(z))}{\exp(z)}-1|) \). The first zero crossing is at z=137.8052295808. For smaller value of z, the error term is dominated by the 1/z laurent term I mentioned earlier; for larger values of z, it is dominated by \( \exp(\exp^{0.5}(\log(z)+2\pi i))+\exp(\exp^{0.5}(\log(z)-2\pi i)) \) Here is the graph. The logarithmic spikes are the zero crossings of f(f(z))-exp(z). Notice how it wobbles around exp(z). The error term is dominated by the error term for f(z), which I graphed in an earlier post. f(z) has nearly the same first error term. f(f(z))/exp(z) has a larger error than f(z)/exp^0.5(z), but they similarly. The error term for f(f(z))/exp(z) at z=1000 is f(f(z))/exp(z)-1=-1.138369235934 E-12.