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

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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],

with Fourier coefficients

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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