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My question is similar to this one. There are ways of deriving the formulae like $$\sum_{k = 1}^\infty \frac{\sin(kz)}{k} = \frac{\pi - z}{2}$$ using summation methods. My question is: How can we argue, without invoking Fourier analysis and preferably using only summation methods, that the formula above holds specifically for $z \in (0, 2\pi)$?

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  • $\begingroup$ It's solved here by using Abel-Plana Formula. $\endgroup$ – Felix Marin Jul 4 '14 at 7:03
  • $\begingroup$ Thanks, but I think you misunderstood my question. $\endgroup$ – glebovg Jul 4 '14 at 7:34
  • $\begingroup$ I just pointed out something that was somehow closed to your question. Those type of question are always interesting. Thanks. $\endgroup$ – Felix Marin Jul 4 '14 at 8:45

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