Gauss's Digamma Theorem

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Theorem

Let $\dfrac p q$ be a positive rational number with $p < q$.


Then:

$\displaystyle \psi \left({\frac p q}\right) = -\gamma - \ln 2 q - \frac \pi 2 \cot \left({\frac p q \pi}\right) + 2 \sum_{n \mathop = 1}^{\left\lceil{q / 2}\right\rceil - 1} \cos \left({\frac {2 \pi p n} q}\right) \ln\left({\sin \left({\frac {\pi n} q}\right)}\right)$

where:

$\psi$ is the digamma function
$\cot$ is the cotangent function
$\ln$ is the natural logarithm.


Proof


Source of Name

This entry was named for Carl Friedrich Gauss.