Determine a unique square root for a holomorphic function on a simply connected set

Question

If we assume that the holomorphic function $f$ is nonzero on a simply connected set A, then we can construct it's logarithm via integral of ${f'\over f}$ along a contour in A, and then we can use this logarithm to construct the square root function of $f$, my question is that if f is zero at some points in A then does this argument still holds? we can define the square root for any complex number but we can't define the logarithm for 0.

Answer 1

Actually, the function that we have to integrate is $\frac{f'}f$ rather than $\frac f{f'}$.

Anyway, the answer is negative in general: if $f\colon\Bbb C\longrightarrow\Bbb C$ is the identity (that is, $f(z)=z$), then $f$ has no holomorphic square root, in spite of the fact that $\Bbb C$ is simply connected.