Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 3 / 3.2A

Table of contents
- a 3.2A
  - Solution a
- b 3.2A
  - Solution b

a

Solution

If

f

is meromorphic inside and on a positively oriented simple closed contour

C

, with no pole on

C

, then

\oint_{C} f (z) d z = 2 πi a inside C \sum Res (f, a) .

b

Solution

Integrate

e^{in z} / (z^{4} + 1)

over the upper semicircle. Jordan's lemma removes the arc and the upper poles are

e^{iπ /4}

and

e^{3 iπ /4}

. Summing their residues and taking real parts gives

\int_{- \infty}^{\infty} \frac{cos ( n x )}{x ^{4} + 1} d x = \frac{π}{2} e^{- n / 2} (cos \frac{n}{2} + sin \frac{n}{2}) .

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