 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2014-resit/10/solution

f_{m}

a_{m}

. Since

x sin (m x)

π a_{m} = 2 \int_{0}^{π} x sin (m x) d x = - \frac{2 π ( - 1 ) ^{m}}{m} .

Consequently the Fourier sine series is

x = 2 n = 1 \sum \infty \frac{( - 1 ) ^{n + 1}}{n} sin (n x), - π < x < π .

The series converges to

x

at each interior point and in the L2 norm. Its periodic extension has jumps at

\pm π

, where the Fourier series instead converges to the midpoint value zero. The overshoot near those jumps is the Gibbs phenomenon.

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