Gauss's Digamma Theorem

Theorem

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

Then:

$\ds \map \psi {\dfrac p q} = -\gamma - \ln 2 q - \dfrac \pi 2 \map \cot {\dfrac p q \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {q / 2} - 1} \map \cos {\dfrac {2 \pi p n} q} \map \ln {\map \sin {\dfrac {\pi n} q} }$

where:

$\psi$ is the digamma function

$\cot$ is the cotangent function

$\ln$ is the natural logarithm.

Proof

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Source of Name

This entry was named for Carl Friedrich Gauss.