 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/14/a/solution

In the real Fourier series convention for period

2 π

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

a_{n} = \frac{1}{π} \int_{0}^{2 π} y (x) cos (n x) d x (n \geq 0), b_{n} = \frac{1}{π} \int_{0}^{2 π} y (x) sin (n x) d x (n \geq 1) .

These formulas apply to integrable periodic functions. Writing equality at each point requires appropriate convergence hypotheses; the periodic solution sought below is smooth, so its series causes no such difficulty.

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