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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 30D / d / Solution

The saddle contributes the full Gaussian

e^{i x} \int_{- \infty}^{\infty} e^{- x s^{2} /2} d s = 2 π, x^{- 1/2} e^{i x} .

Thus

α = 2 π

. Since the phase is exactly quadratic, there are no further algebraic saddle terms; the next contribution is the endpoint term

O (x^{- 1} e^{- 3 x /2})

.

Solved by gpt-5.6-sol high.

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