closed form for regular superfunction expressed as a periodic function

[sheldonison]

Or it does already, but just a wee bit faster than exponential. This makes me wonder about an interesting place for mathematical exploration: the behavior of entire functions given by a Taylor series whose terms' coefficients decay just a "wee" bit faster than exponential. As this example shows, such functions can have extremely complicated behavior (note the complicated "fractal structure" of the graphs of these superfunctions.).

Yes, very interesting. I notice that the fractals are often very sparse too. Its only growing super-exponentially on a filagree, and most of the rest of the function is not growing nearly as fast. So it looks like spike singularities. Not sure if that helps any.

I think I may have also figured out the closed forms for the abel functions, the inverse superexponential developed from the fixed point, which I'll eventually post, when I have time to verify the equations.

Now I'm stuck on eta, \( \eta=e^{(1/e)} \). I'm trying the substitution y=1/z. I think my results so far are bogus though, so I'm editing them out.
-Sheldon