Derivative of exp^[1/2] at the fixed point?

[sheldonison]

Is the derivative is also continuous at this singularity, and if so, then what is its value? If the derivative is continuous at the singularity, how many of the higher derivatives are also continuous?

In particular, there is a formal half iterate that is not real valued that can be developed at the fixed point L.
\( \exp(L+z) = L \cdot (1 + z + \frac{z^2}{2} + \frac{z^3}{6} + ... \)

And the formal half iterate begins with
\( \exp^{0.5}(z+L) = L + \sqrt{L}z + \frac{L \cdot z^2}{2(L+\sqrt{L})} + ... \)

Numerical experiments suggest that the derivative of Knesser's real valued half iterate is continuous at L, and the first derivative at the singularity at L is \( \sqrt{L} \), and that the 2nd derivative may also match the formal half iterate.