 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/17/a/solution

With the usual real normalization, the Fourier series is

f (x) \sim \frac{a _{0}}{2} + \sum_{n = 1}^{\infty} [a_{n} cos n x + b_{n} sin n x],

a_{n} = \frac{1}{π} \int_{- π}^{π} f (x) cos n x d x (n \geq 0), b_{n} = \frac{1}{π} \int_{- π}^{π} f (x) sin n x d x (n \geq 1) .

These formulas follow from Fourier orthogonality. Integrability is needed to define the coefficients, and periodicity alone does not guarantee pointwise equality with the series. For a piecewise smooth periodic function, the Dirichlet-Jordan convergence theorem gives the average of the one-sided limits at a jump and the function value at a continuity point.

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