 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-2/18/b/iii/solution

Write the translated Fourier series in the convention

a_{0}^{(g)} /2 + \sum_{n \geq 1} (a_{n}^{(g)} cos k_{n} x + b_{n}^{(g)} sin k_{n} x)

. Comparing with (i) gives

a_{0}^{(g)} = 0, a_{n}^{(g)} = - b_{n} sin \frac{2 πn ℓ}{L}, b_{n}^{(g)} = b_{n} cos \frac{2 πn ℓ}{L} .

The minus sign in the cosine coefficient records translation in the positive

x

direction.

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