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If one sees the simplification done in equation $5.3$ (bottom of page 29) of this paper it seems that a trigonometric identity has been invoked of the kind,

$\ln(2) + \sum _ {n=1} ^{\infty} \frac{\cos(n\theta)}{n} = - \ln \left\vert \sin\left(\frac{\theta}{2}\right)\right\vert $

Is the above true and if yes then can someone help me prove it?

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    Are you familiar with writing $\cos$ as a sum of complex exponentials? Are you familiar with the Taylor series expansion of $-\log(1-x)$?2012-07-24

1 Answers 1

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Hint 1: Use that $ \log(1-z)=-\sum\limits_{n=1}^\infty\frac{z^n}{n} $

Hint 2: Set $ z=r e^{i\theta} $

Hint 3: Take a real part

Hint 4: Take a limit $r\to 1-0$ and use Abel's summation formula

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    @6818, you still need to take the series route...2012-07-25