 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-336/1/b/solution

The exponential phase is

Φ (s) = - s + 2 λ - \frac{λ ^{2}}{s} .

Φ^{'} (s) = - 1 + \frac{λ ^{2}}{s ^{2}} = 0, s = \pm λ .

For positive real

λ

, the original positive-real contour passes through

s = λ

s = 0

separates it from the other saddle. At the contributing saddle,

Φ (λ) = 0, Φ^{''} (λ) = - \frac{2}{λ} .

s = λ + λ^{1/2} t

. Then

Φ (s) = - t^{2} + O (λ^{- 1/2} t^{3}), d s = λ^{1/2} d t .

I (λ) \sim λ^{- 1/2} λ^{1/2} \int_{- \infty}^{\infty} e^{- t^{2}} d t = π .

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