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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!

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    If $f=k$ and $f$ doesn't vanish then it is true that $f'/f = k'/k$, simply because $f=k$ and $f'=k'$; no logs needed. It is also true that whenever a logarithm of $f$ exists, i.e. a function $p$ such that $\exp(p)=f$, then it is true that $p'=f'/f$, a consequence of the chain rule. I'm not sure exactly what the question is here. Maybe you should ask the person who asked you for clarification of what they are asking. They might just be hinting at the fact that logarithms don't always exist globally and aren't unique, so care should be taken in what "log" means.2011-12-04
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    One thing to note: There is no continuous logarithm defined on the range of a nonconstant entire function. However, for every nonvanishing analytic $f:U\to \mathbb C$, where $U$ is simply connected, there exists an analytic $p:U\to \mathbb C$ such that $\exp(p)=f$; such a $p$ is called a logarithm of $f$.2011-12-04

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