I'm a little confused about the branch cut thing. Given an entire functions $f(z),g(z),h(z)$, $z\in \mathbb C$, such that $f(x)=g(x)+h(x)$ for all $x\in \mathbb R$, $f$ and $g+h$ doesn't vanish on $\mathbb R$ . I take the $\log$ for both sides then differentiate and get:
$$\log f(x)=\log(g(x)+h(x))$$ $$\frac{f'(x)}{f(x)}=\frac{g'(x)+h'(x)}{g(x)+h(x)}$$
I thought this is correct, but I was asked about "what is the branch cut I used here?", and I don't know what does this mean, and what is the answer for this question! Any help!?
Edit: I also have the same problem with the following case:
If $f(x)=e^{g(x)}$ then $\log f(x)=g(x)$, and $\frac{f'(x)}{f(x)}=g'(x)$, also a branch cut issue!