 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2014-resit/10/a/ii/solution

⟨ 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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