 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/17/b/solution

The Fourier coefficients follow by multiplying the expansion by

1

, by the appropriate cosine, or by the appropriate sine, then using orthogonality. In the convention with constant term

a_{0} /2

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

The factor

1/ L

comes from the squared norms of the nonconstant sine and cosine functions: the integral of each square over

[- L, L]

L

. The constant function has squared norm

2 L

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