Problem. By using the Fourier series expansion of f(x)=cosαx, x∈(−π,π), prove that cotx=1x+∞∑n=1(1x−nπ+1x+nπ)=1x+∞∑n=12xx2−n2π2 and also cscx=1x+∞∑n=1(−1)n(1x−nπ+1x+nπ)=1x+∞∑n=1(−1)n2xx2−n2π2. Also evaluate ∞∑n=11n2−α2.The surprise should be: there is apparently no evidence that Fourier series is involved!
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:
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