 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ii/paper-3/14a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

For

Re k > 0

, choose the straight path from

- i

i

and put

t = i s

- 1 \leq s \leq 1

. Choose the power of

1 - s^{2}

continuously, positive for real positive

k

. The boundary factor vanishes at both endpoints. At zero,

w (0) = i \int_{- 1}^{1} (1 - s^{2})^{k - 1} d s = i \int_{0}^{1} u^{- 1/2} (1 - u)^{k - 1} d u = i B (k, 1/2),

using the symmetric beta function integral. This choice also gives a nonzero solution.

 New to topics? Read the docs here!