Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 10 / a / ii

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- Solution ii

Solution

The product-to-sum formula gives

⟨ f_{n}, f_{m} ⟩ = \frac{1}{2} \int_{- π}^{π} [cos ((n - m) x) - cos ((n + m) x)] d x = 0 (n \neq = m) .

When both indices are positive the two integer frequencies integrate to zero. If either index is zero the integrand already vanishes. This is Fourier orthogonality.

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