Find a single-valued analytic branch of $\sqrt{z^2-1}$ in $\mathbb{C} \backslash [-1,1]$.

complex-analysis complex-numbers

Solution 1:

EDIT: A more straightforward solution is to note that $z^2 - 1$ is in $(-\infty,0]$ exactly when $z$ is in the interval $[-1,1]$ or the imaginary axis. If we use the principal branch of $\sqrt{}$, $\sqrt{z^2-1}$ will be analytic everywhere else. This would be positive on $[1,\infty)$, so instead we take $-\sqrt{z^2-1}$. To avoid having a branch cut on the imaginary axis, we switch to $+\sqrt{z^2-1}$ in the left half plane. Thus $$f(z) = \cases{-\sqrt{z^2-1} & for $\text{Re}(z) \ge 0$\cr +\sqrt{z^2-1} & for $\text{Re}(z) < 0$\cr}$$

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