 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-29/2/d/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

Substitute

R = r^{π / α}

P_{1} {B [0, T (r)] \subset W_{α}} = \frac{4}{π} arctan (r^{- π / α}) = \frac{2}{π} arctan (\frac{2 r ^{π / α}}{r ^{2 π / α} - 1}) .

It tends to

1

r ↓ 1

, and is asymptotic to

(4/ π) r^{- π / α}

r \to \infty

. The exponent reflects how the conformal map stretches wedge angles; wider wedges give slower decay.

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