Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 14E / b

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- Solution b

Solution

At a nonpositive integer, the branch disappears and the collapsed Hankel contour extracts the residue at zero. For

s = 0

,

β (0) = Res_{t = 0} \frac{1}{t ( e ^{t} + e ^{- t} )} = \frac{1}{2} .

For

s = - 2

, the coefficient of

t^{- 1}

in

t^{- 3} (\frac{1}{2} - \frac{1}{4} t^{2} + O (t^{4}))

is

- 1/4

; multiplication by

Γ (3) = 2

gives

β (- 2) = - \frac{1}{2} .

When

s = - 2 n - 1

, the integrand is

t^{- 2 n - 2}

times an even power series, so it has no

t^{- 1}

coefficient. Hence

β (- 2 n - 1) = 0 (n \geq 0) .

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