 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2012/s1/2/a/solution

f (z) = z + z^{- 1}

C ∖ {0}

. A holomorphic function is locally conformal where its derivative is nonzero. Here

f^{'} (z) = 1 - z^{- 2}

vanishes at

z = \pm 1

. Therefore

holomorphic on C ∖ {0}; conformal at every point except 0, 1, - 1,

where the exclusion of zero is because the function is undefined there. Holomorphy alone does not supply conformality at the two critical points.

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