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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 7E / a / Solution

The partial-fraction decomposition is

\frac{1}{1 + t ^{2}} = \frac{1}{2 i} (\frac{1}{t - i} - \frac{1}{t + i}) .

The residues at

i

and

- i

are respectively

1/ (2 i)

and

- 1/ (2 i)

. If two paths from

0

to

z

have winding-number differences

n_{+}

and

n_{-}

about those poles, their integrals differ by

2 πi (\frac{n _{+}}{2 i} - \frac{n _{-}}{2 i}) = π (n_{+} - n_{-}) .

Loops can wind around either pole independently, so the value changes by arbitrary integer multiples of

π

and

F

is multivalued.

Along the straight real segment,

\int_{0}^{1} \frac{d t}{1 + t ^{2}} = \frac{π}{4} .

Consequently the complete set of values is

F (1) = \frac{π}{4} + kπ, k \in Z .

Solved by gpt-5.6-sol high.

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