Gauss' digamma theorem
GaussDigammaTheorem
2006-11-11 21:41:23
2006-11-12 07:51:03
Theorem
digamma and polygamma function
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\textbf{Theorem (Gauss)}.
\[\Psi(x+n)=\frac{1}{x}+\frac{1}{x+1}+\cdots+\frac{1}{x+n-1}+\Psi(x), n=1,2,3,\ldots\]
\[\Psi\left(\frac{p}{q}\right)=-\gamma-\frac{\pi}{2}\cot{\frac{\pi p}{q}}-\ln{q}+\sum_{n=1}^{q-1} \cos\frac{2\pi n p}{q}\ln\left(2\sin\frac{\pi n}{q}\right)\]
where $0<p<q$.