I have this HW and I don't know how to approach it, does anybody know how can it be shown that $$∑_{n=1}^∞\frac{1}{n^2} =\frac{π^2}{6}$$ using residue theorem?
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2This is not a physics question. It belongs on [math.se] – Bill N Dec 25 '17 at 20:35
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1Complex Variables - Schaum's Outline Series by Spiegel-Lipschutz S.-Schiller-Spellman – Frobenius Dec 25 '17 at 22:14
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1It has been done by tired here and this thread is a must. – Jack D'Aurizio Dec 26 '17 at 00:57
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Can you apply the residue theorem to $\int_{|z| = 2\pi (N+1/2)} \frac{z^{-2}}{e^z-1}dz$ ? Do you see why $\lim_{N \to \infty}\int_{|z| = 2\pi (N+1/2)} \frac{z^{-2}}{e^z-1}dz= 0$ ? – reuns Dec 26 '17 at 09:21
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In four dense pages in this paper the author starts with the residue theorem on page 7 and proves that $\zeta(2)=\pi^2/6.$ I skimmed the article and it looks like a clear exposition to me. There are certainly other versions.
The paper is by Brendan Sullivan, Numerous Proofs of $\zeta(2)=\frac{\pi^2}{6},$ dated 2013.
daniel
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@yasiren: no problem. actually the proof is shorter than i thought--maybe two and a half pages, and nothing very complicated. – daniel Dec 25 '17 at 20:44
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I cannot imagine what satisfaction people get from doing other people's homework for them. – WillO Dec 25 '17 at 23:21
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@WillO: Neither my ref. nor those in the comments will do the OP's HW for him/her, and I don't assume bad faith on the part of the questioner. – daniel Dec 27 '17 at 09:27