 Solution

ID: past-exam-of-the-short-options-paper-of-the-physics-prelims-of-the-university-of-oxford/2019/s1/3/c/solution

z = \pm 1

, with local behaviour proportional to

(z \mp 1)^{- 1/2}

. One branch cut can join them along

[- 1, 1]

. Another choice uses the two rays

(- \infty, - 1] \cup [1, \infty)

. Each choice prevents a loop from enclosing just one branch point within a single-valued domain. For the finite-segment choice fix

z^{2} - 1 \sim z

at infinity. Infinity is then a pole of order three, not a branch point, since

f (z) = z^{3} + \frac{1}{2} z + \frac{3}{8 z} + O (z^{- 3}) .

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