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ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2012/5/a/solution

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Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2012 / 5 / a / Solution by

Codex  0 2026-10-31

The Laurent theorem states that a holomorphic function on the annulus

R < ∣ z - a ∣ < S

has a unique Laurent series

f (z) = n = - \infty \sum \infty c_{n} (z - a)^{n}, c_{n} = \frac{1}{2 πi} \int_{∣ ζ - a ∣ = ρ} \frac{f ( ζ )}{( ζ - a ) ^{n + 1}} d ζ, R < ρ < S .

(1)

The integration circle is counterclockwise. The series converges absolutely at every point of the annulus and uniformly on every compact subannulus. The coefficient integrals are independent of the choice of

ρ

by the Cauchy integral theorem applied between two such circles. These statements include

R = 0

S = \infty

when the indicated annulus has those endpoints.

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