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Sunday, December 15, 2013

Fourier Series

Usual exercises in Fourier series ask students to prove identity of the form f=ancosnx+bnsinnx. Most of them are extremely straightforward and what you need is sufficient manpower, patience and time. If I were the one who teach Fourier series, I would propose the following identities as examples:
Problem. By using the Fourier series expansion of f(x)=cosαx, x(π,π), prove that cotx=1x+n=1(1xnπ+1x+nπ)=1x+n=12xx2n2π2 and also cscx=1x+n=1(1)n(1xnπ+1x+nπ)=1x+n=1(1)n2xx2n2π2. Also evaluate n=11n2α2.
The surprise should be: there is apparently no evidence that Fourier series is involved!

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