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Here $\psi(z)$ is digamma function, $\Gamma(z)$ is gamma function. $$\psi(z)=\frac{{\Gamma}'(z)}{\Gamma(z)},$$ For positive integers $m$ and $k$ (with $m < k$), the digamma function may be expressed in terms of elementary functions as: $$\psi\left(\frac{m}{k}\right)=-\gamma-\ln(2k)-\frac{\pi}{2}\cot\left(\frac{m\pi}{k}\right)+2\sum^{[(k-1)/2]}_{n=1}\cos\left(\frac{2\pi nm}{k}\right)\ln\left(\sin \left(\frac{n\pi}{k}\right)\right). $$ How to prove it ?

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You can look at this, and the references therein.

Added: In fact, a quick Google search gives several references for the proof. Also, if the math does not render well, the Planetmath team suggests to switch the view style to HTML with pictures (you can choose at the bottom of the page).

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  • $\begingroup$ @M Turgeon Thank you very much! I think it's helpful, but I can't find a simple proof. $\endgroup$
    – Daoyi Peng
    May 22, 2012 at 4:21
  • $\begingroup$ @DaoyiPeng What would be a simple proof for you? $\endgroup$
    – M Turgeon
    May 22, 2012 at 12:29
  • $\begingroup$ The proof is ill formatted! $\endgroup$
    – Pedro
    May 22, 2012 at 23:21
  • $\begingroup$ @PeterTamaroff Well, I sent a comment so that someone check and fix it. Meanwhile, it is still possible to look at the source file and figure out what is not being processed. $\endgroup$
    – M Turgeon
    May 23, 2012 at 0:27
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    $\begingroup$ @MTurgeon Thanks! $\endgroup$
    – Pedro
    May 28, 2012 at 20:19

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