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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/4/a/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 4 / a / Solution by

Codex  0 2026-10-18

A sufficient contour setting for both results is a positively oriented, simple, closed, piecewise smooth contour

C

, with the function holomorphic on an open neighbourhood of

C

and its entire interior. The Cauchy integral theorem then states

\oint_{C} f (z) d z = 0 .

If

z_{0}

is strictly inside

C

, the Cauchy integral formula states

f (z_{0}) = \frac{1}{2 πi} \oint_{C} \frac{f ( z )}{z - z _{0}} d z .

The contour must not pass through

z_{0}

. Positive orientation means anticlockwise; reversing it reverses the sign of the contour integral. The holomorphy assumption throughout the interior excludes hidden singularities there.

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