Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 312 / 3 / ii

Solution

F_{2} (q, k - q) = O (k^{2} / q^{2})

for fixed

k

q \to \infty

. In this limit the two hard vectors are opposite,

\frac{q \cdot ( k - q )}{q ∣ k - q ∣} \to - 1, \frac{∣ k - q ∣}{q} + \frac{q}{∣ k - q ∣} \to 2.

The constant part of the stated kernel is therefore

\frac{5}{7} - 2 A + \frac{2}{7} = 1 - 2 A .

It must vanish, giving

A = \frac{1}{2} .

F_{2}

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