Digamma Function - Gauss's Digamma Theorem

Gauss's Digamma Theorem

For positive integers m and k (with m < k), the digamma function may be expressed in finite many terms of elementary functions as

$\psi\left(\frac{m}{k}\right) = -\gamma -\ln(2k) -\frac{\pi}{2}\cot\left(\frac{m\pi}{k}\right) +2\sum_{n=1}^{\lfloor (k-1)/2\rfloor} \cos\left(\frac{2\pi nm}{k} \right) \ln\left(\sin\left(\frac{n\pi}{k}\right)\right)$

and because of its recurrence equation for all rational arguments.

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