 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-203/2/b/solution

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

g_{K} (i y) = u_{y} + i v_{y}

u_{y} = O (y^{- 2}), v_{y} = y - \frac{a _{K}}{y} + O (y^{- 2}), \frac{v _{y}}{y} \to 1.

The real interval under consideration lies outside the hull, so the reflected boundary map sends it monotonically to

(g_{K} (x), g_{K} (b))

P_{i y} (B_{T} \in (x, b)) = \frac{1}{π} \int_{g_{K} (x)}^{g_{K} (b)} \frac{v _{y}}{( r - u _{y} ) ^{2} + v _{y}^{2}} d r .

The endpoints are fixed and finite. On this interval,

\frac{y v _{y}}{( r - u _{y} ) ^{2} + v _{y}^{2}} ⟶ 1

uniformly, using

v_{y} / y \to 1

and

u_{y} \to 0

. Integrating gives the interval-length limit:

y \to \infty lim π y P_{i y} (B_{T} \in (x, b)) = g_{K} (b) - g_{K} (x) .

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