The upper superexponential

[sheldonison]

\( F_{4,3} \) is entire, has period \( 2\pi i/\ln(2\ln(2)) \).
\( F_{2,3} \) is not entire, has period \( 2\pi i/\ln(\ln(2)) \).
....
They have no limit along the imaginary axis because they are imaginary periodic.

When I first looked at Dimitrii's graphs in "Bummer", I didn't realize that the two functions were completely different functions in the imaginary plane, and have different imaginary periods! What I noticed was one had cut points, and the other had fractal behavior. Are the imaginary periods exactly repeating copies?

Functions \( F \) above are related: \( F_{2,3} \) can be expressed through \( F_{2,1} \) and \( F_{4,3} \) can be expressed through \( F_{4,5} \) with some complex constant ofsets of the arguments.

The fractal behavior of \( F_{4,3} \) is \( F_{4,5} \) increasing to infinity via tetration, except it is occurring at the i=imaginary_period/2 line, with real values! But otherwise, the fractal behavior is as one would expect! It sounds as though the conversions are as simple as:

\( F_{2,1}(z)=F_{2,3}(z+\text{complexoffset1}) \)
\( F_{4,5}(z)=F_{4,3}(z+\text{complexoffset2}) \)
\( F_{2,3}(z)= F_{4,3}(z+\theta(z)) \),

Where the complex offset is just a real offset plus half of the imaginary period of each function.
This means \( \theta \) along with the complex offsets, also allows conversions between \( F_{2,1} \) and \( F_{4,5} \), the lower superexponential, and the upper superexponential.