# few short question about branch-cut and branch points

Q. Find branch points and construct branch lines for the functions
$\displaystyle (a) f(z) = \left( \frac{z}{1-z} \right)^{\frac 1 2 }$
$\displaystyle (b) \left( z^2 - 4\right)^{\frac 1 3 }$
$\displaystyle (c) f(z) = \ln (z-z^2)$

(I am guessing) For $(a)$ the branch point is $0$ and $\infty$ and the branch like will be any straight line extending from $0$ to $\infty$, for $(b)$ the branch points are $\pm 2$ and branch line would be a line joining these two points, and for $(c)$ the branch point are $0$ and $1$ and I don't know how to draw branch line here.

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I have no time for a proper answer, but your guess for (a) isn't right: Notice that the fraction goes to $-1$ at infinity, so that is not a problem from a branching point of view. $z=1$, on the other hand, … – Harald Hanche-Olsen Mar 1 '13 at 12:05
so $z=1$ is the branch point? – hasExams Mar 1 '13 at 12:26
Yes, and $z=0$ also. As for (b) and (c), watch out for infinity. Think about what happens when $|z|$ is very large: In (b) you almost get $z^{4/3}$, and in (c) you almost get $\ln(-z^2)$. (For “almost” read “asymptotically”, really.) – Harald Hanche-Olsen Mar 1 '13 at 12:52