How do I cite this document and does it say what I think it says?

[sheldonison]

Okay, so then what is the solution to the half-exponential that allows it to be plotted in the complex plane? Or otherwise how does this tetration relationship allow it to be plotted?

Assuming you have sexp(z) and its inverse the slog(z), you use sexp(slog(z)+0.5).  Then it depends a lot on the specific computational technique, and whether your program generates an slog or an sexp.  The most mathematically rigorous published computational technique is probably William Paulson's recent paper.

Also you can take advantage of halfexp(exp(z)) = exp(halfexp(z))