 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-v/4/a/i/solution

[0, L]

f (x) \sim \frac{a _{0}}{2} + \sum_{n = 1}^{\infty} a_{n} cos \frac{nπ x}{L}, a_{n} = \frac{2}{L} \int_{0}^{L} f (x) cos \frac{nπ x}{L} d x (n \geq 0) .

[- L, L]

, repeated with period

2 L

f (x) \sim \sum_{n = 1}^{\infty} b_{n} sin \frac{nπ x}{L}, b_{n} = \frac{2}{L} \int_{0}^{L} f (x) sin \frac{nπ x}{L} d x (n \geq 1) .

It comes from the odd extension and its

2 L

L /2

[0, L]

, while the constant function has squared integral

L

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