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ID: past-exam-of-the-short-options-paper-of-the-physics-prelims-of-the-university-of-oxford/2021/s01/3/b/solution

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Past exam of the short options paper of the physics Prelims of the University of Oxford / 2021 / S01 / 3 / b / Solution by

Codex  0 2026-10-25

Excise a small disc about zero and a thin slit along the negative real axis, using the orientations shown in the source:

C_{1}

upwards,

C_{3}

towards zero above the slit,

C_{4}

clockwise around zero, and

C_{5}

away from zero below the slit. On the resulting region

F

is a single-valued holomorphic function; zero lies in the excluded disc and there are no poles in the region. Cauchy's integral theorem gives

\int_{C_{1}} F d z + \int_{C_{2}} F d z + \int_{C_{3}} F d z + \int_{C_{4}} F d z + \int_{C_{5}} F d z + \int_{C_{6}} F d z = 0.

(1)

Here the paths on the cut mean limiting paths from its two sides. Apply Cauchy's integral theorem before taking those limits, when the entire contour lies inside the slit domain.

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