 Solution

ID: past-exam-of-the-short-options-paper-of-the-physics-prelims-of-the-university-of-oxford/2021/s01/1/c/solution

Assume positive, counterclockwise orientation of a contour. Factor the polynomial denominator as

z^{4} - 10 z^{2} + 1 = (z^{2} - (5 - 26)) (z^{2} - (5 + 26)) .

Only

z = \pm (3 - 2)

P (z)

, their residues are

\frac{z}{P ^{'} ( z )} = \frac{1}{4 z ^{2} - 20} = - \frac{1}{8 6} .

The residue theorem therefore gives

I = 2 πi (- \frac{1}{4 6}) = - \frac{πi}{2 6} .

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