 Solution

ID: past-exam-of-the-short-options-paper-of-the-physics-prelims-of-the-university-of-oxford/2021/s01/2/b/solution

The poles are the four roots of

z^{4} = - 1

. The upper half-plane contains

a = e^{iπ /4}

and

b = e^{3 iπ /4}

, both enclosed when

R > 1

q

Res (\frac{z ^{2}}{z ^{4} + 1}, q) = \frac{q ^{2}}{4 q ^{3}} = \frac{1}{4 q} .

The sum of the two residues is

\frac{1}{4} (e^{- iπ /4} + e^{- 3 iπ /4}) = - \frac{i}{2 2} .

The residue theorem, with the real segment traversed from

- R

R

and the upper arc from

R

- R

, yields

\oint_{Γ_{R}} \frac{z ^{2}}{z ^{4} + 1} d z = \frac{π}{2} .

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