 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ib/paper-1/12/12-1g/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

∣ ζ ∣ = ρ

with

r < ρ < R

, Cauchy's formula applied inside and outside the circle gives the locally uniformly convergent Laurent expansion

f (z) = \sum_{n = - \infty}^{\infty} a_{n} z^{n}, a_{n} = \frac{1}{2 πi} \oint_{∣ ζ ∣ = ρ} \frac{f ( ζ )}{ζ ^{n + 1}} d ζ .

At zero, the singularity is removable exactly when all

a_{n}

with

n < 0

vanish; it is a pole of order

k

when

a_{- k} \neq = 0

and

a_{n} = 0

for

n < - k

; it is essential when infinitely many negative coefficients are nonzero.

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