Searching for an asymptotic to exp[0.5]

[sheldonison]

....
f(x) is the half-iterate from the sinh method.

I assume since sinh is an odd function, that f(x), the asymptotic half iterate, would also be an odd function, though this is not required. As I remember, the closest singularity to the origin for sinh^{0.5} is on the imaginary axis.

For f(x) asymptotic to exp^{0.5}, the branch cut is on the negative real axis. For the asymptotic to sinh^{0.5}, one possible branch cut is on the imaginary axis, in both directions, which would lead to an odd function, with even Taylor series coefficients=0. Is this what you had in mind?