I am asked to find the principle branch of the complex function $$f(z) = \sqrt{1-z}$$
I know that the principle branch of $z^{1/2}$ is given by $\exp(\frac{1}{2} \log(z))$ where $$\log(z) = \log(|z|)+i\arg(z)$$ is the principle branch of the logarithm. So then to find the principle branch of $\sqrt{1-z}$, is it just
$$\exp\bigg(\frac{1}{2} \log(1-z)\bigg)$$
with $\log(1-z)$ being the principle branch of the logarithm i.e. $$\log(1-z) = \log(|1-z|) + i\arg(1-z).$$
I think I am confusing myself. Is the principle branch correct?
Best Answer
You need to specify what you do with the arg(1-z) when $z$ winds around 1 (and the arg increases with $2\pi$). You will have to introduce a cut (where arg, whence $\sqrt{1-z}$ is not defined), a typical choice here being $[1,+\infty[$ along the positive real axis. This is compatible with the principal branch of Log(w) which is usually defined by a cut along $R_-$.