An explanation for this?

[tommy1729]

For base e exponent, as far as I know, the only way to generate the half iterate (or any other partial iterate) is indirectly, via \( \text{sexp}(\text{slog}(z)+0.5) \). I'm unaware of any other way to generate the power series.
- Shel

you could also use my method since e > e^(1/2).

although its not directly a power series nor the best numerically method its simple and doesnt use slog or sexp.

regards

tommy1729