Prove that one can define a branch of the function $\sqrt{1-z^2}$ in every region $D\subset \mathbb{C}$ such that the points $-1$ and 1 belong to the same connected component of the complement of $D.$ How many values can the integral $$\int_\gamma \frac{\mathrm{d}z}{\sqrt{1-z^2}}$$ take along a closed path $\gamma$ contained in $D$ ?
2026-03-26 10:45:12.1774521912
possible results of integral along closed path after defining branch of sqrt
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