# Branch cut of the principle value log

#### bmxicle

1. Homework Statement
Find the branch points of g(z) = log(z(z+1)/(z-1)) and defining a branch of g as the principle branch of the logarithm find the location of the branch cuts.

2. Homework Equations

3. The Attempt at a Solution

Since $$g(z) = log(z) + log(z+1) - log(z-1)$$ the branch points are 0, 1, -1 and infinity.

To find the branch cuts we need to find where the argument is $$arg(z) \in (-\pi,\pi]$$

putting each branch point into polar coordinates gives:
$$z1 = r_1e^{i\theta_1}$$
$$z2 = r_2e^{i\theta_2} -1$$
$$z3 = r_3e^{i\theta_3} +1$$
$$z = ln(\left|{\dfrac{r_1 r_2}{r_3}}\right|) +i(\theta_1 + \theta_2- \theta_3)$$

let $$c = ln(\left|{\dfrac{r_1 r_2}{r_3}}\right|)$$ and then:

$$(-\infty, -1) \Rightarrow g(z) = c + \pi i$$
$$(-1, 0) \Rightarrow g(z) = c + 2\pi i$$
$$(0, 1) \Rightarrow g(z) = c + \pi i$$
$$(1, \infty) \Rightarrow g(z) = c$$

So this gives $$(0, 1)$$ as the point where the branch cut isn't defined, but the solution I was given says that $$(1, \infty)$$ is also not defined, so I'm not sure where I'm going wrong.

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