Analyze the Complex Function by using the Principal log Branch I am trying to analyze the function $\sqrt{1-z^2}$, where the square root function is defined by the principal branch of the log function. I want to locate the the discontinuities.
I know the discontinuities will lie on the negative real axis but I cannot figure out for which values of $z$ this will occur. I first tried rewriting the function as $(1-e^{2i\theta})$. But this seems to be getting me nowhere. Any thoughts?

$$f(z)=\sqrt{1-z^2}=\left(1-z^2\right)^{1/2}=e^{\log\left(1-z^2\right)/2}\,.$$
The branch cut of $\log$ runs from zero to negative infinity, therefore $f$ is discontinuous at point $z$ whenever $1-z^2\in{({{-\infty};{0}}]}$, hence the solution set of $$1-z^2\in{({{-\infty};{0}}]}$$ for $z$ over $\mathbb{C}$ will be the set of points of discontinuity of $f$.
