Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 7B (Further Complex Methods)

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- Solution 7B

Solution

Integration by parts is legitimate after symmetrically excluding a neighborhood of zero; the boundary terms vanish because

cos (n x) - cos (m x) = O (x^{2})

at zero and is bounded at infinity. Hence

PV \int_{- \infty}^{\infty} \frac{c o s n x - c o s m x}{x ^{2}} d x = - n PV \int_{- \infty}^{\infty} \frac{s i n n x}{x} d x + m PV \int_{- \infty}^{\infty} \frac{s i n m x}{x} d x .

The standard Dirichlet integral, obtained by closing the contour for

e^{ia z} / z

in the upper half-plane and indenting above its pole at zero, is

PV \int_{- \infty}^{\infty} \frac{s i n ( a x )}{x} d x = π (a > 0) .

Therefore

PV \int_{- \infty}^{\infty} \frac{cos n x - cos m x}{x ^{2}} d x = π (m - n) .

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