I want to determine if $z = 1$ is a branch point of $\log(\frac{z – 1}{z+1})$. I think it should be a branch point because we would have $\log(0)$ , yet somehow my work shows that it isn't, so I'm wondering if I've done something wrong.
Write $z – 1 = R_1e^{i \theta_1}$ and $z + 1 = R_2e^{i \theta_2}$. As we encircle the point $z = 1$, moving around the circle, $\theta_1$ changes by $2\pi$ and $\theta_2$ changes by $0$. Thus,
$$\log(\frac{z-1}{z+1}) = \log(\frac{R_1}{R_2}e^{i(\theta_1 – \theta_2)})$$
and at the start, we are at $\log(\frac{R_1}{R_2}e^0)$, while after encircling $z = 1$, we are at $\log(\frac{R_1}{R_2}e^{i(2\pi – 0)})$. $e^{i 2\pi} = \cos(2\pi) + i\sin(2 \pi) = 1$, so after encircling, the function value does not change, hence $z = 1$ is not a branch point.
Is this correct reasoning? if not, where was my error?
Best Answer
The source of the error is this sentence:
It is not true that just because the interior terms do not change, the function value does not change. In fact, this is not true for any complex $z$, since $\log$ is multi-valued. At the start, the interior term is indeed at $\frac{R_1}{R_2}e^0$, and at the end, the interior term is indeed at $\frac{R_1}{R_2}e^{i(2\pi - 0)}$, but it says nothing about the behavior of the complex log at those points. Nothing concrete can be said about the entire function until we write
$$\log(\frac{R_1}{R_2}e^{i(\theta_1 - \theta_2)}) = \operatorname{Log}(\frac{R_1}{R_2}) + i(\theta_1 - \theta_2)$$
where $\operatorname{Log}$ denotes the real-valued logarithm, and now it is obvious that $z = 1$ would be a branch point.