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ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2005/4/iii/solution

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Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2005 / 4 / iii / Solution by

Codex  0 2026-10-31

Take the unit circle

γ (t) = e^{i t}

0 \leq t \leq 2 π

, in the punctured complex plane. Direct parameter integration gives

\int_{γ} \frac{d z}{z} = \int_{0}^{2 π} e^{- i t} i e^{i t} d t = 2 πi \neq = 0.

(1)

A global holomorphic antiderivative would make this integral zero by the previous part. Thus there is no such antiderivative on the punctured plane. Local primitives can exist without combining into one single-valued function on this domain.

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