Past exam of the metric spaces and complex analysis course of the University of Oxford / 2017 / 5 / a / i

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- Solution i

Solution

Laurent's theorem states that a function holomorphic on

R < ∣ z - a ∣ < S

has a unique expansion

f (z) = \sum_{n = - \infty}^{\infty} c_{n} (z - a)^{n},

converging uniformly on every compact subannulus. Its coefficients are

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

with counterclockwise orientation. The value is independent of the chosen intermediate radius. The nonnegative powers form a power series convergent inside the outer radius, and the negative powers form the principal part of a Laurent series.

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