Complex Methods | Part IB, 2005

Determine the Fourier expansion of the function f(x)=sinλxf(x)=\sin \lambda x, where πxπ-\pi \leqslant x \leqslant \pi, in the two cases where λ\lambda is an integer and λ\lambda is a real non-integer.

Using the Parseval identity in the case λ=12\lambda=\frac{1}{2}, find an explicit expression for the sum

n=1n2(4n21)2\sum_{n=1}^{\infty} \frac{n^{2}}{\left(4 n^{2}-1\right)^{2}}

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