sqrt(exp)

[bo198214]

I do not expect any other singularities. There are only 2 branchpoints and only two cutlines.

Hm perhaps then they lie on other branches.
Those singularities need not to be branch points, they can also be isolated singularities.

But why do there have to be singularities?
First one would expect that a half iterate has the same fixed points as the function. (Is this true for your branch of \( \exp^{1/2} \), i.e. is \( L_2 \) a fixed point?)
But if so, a non-integer iterate can usually be holomorphic at most at one fixed point. This is the case for regular iteration at that fixed point. As your \( \exp^{1/2} \) is not regular at any fixed point (regular iteration yields non-real values at the real axis for \( \exp \)) it must be singular at any fixed point. However if it is non-singular at \( L_2 \) then this may be due to \( L_2 \) being not a fixed point of \( \exp^{1/2} \) in the branch that you chose.
Can you please verify this?