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Gauss' digamma theorem
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(Theorem)
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Theorem (Gauss). $$ \Psi(x+n)=\frac{1}{x}+\frac{1}{x+1}+\cdots+\frac{1}{x+n-1}+\Psi(x), n=1,2,3,\ldot $$ $$ \Psi\left(\frac{p}{q}\right)=-\gamma-\frac{\pi}{2}\cot{\frac{\pi p}{q}}-\ln{q}+\sum_{n=1}^{q-1} \cos\frac{2\pi n p}{q}\ln\left(2\sin\frac{\pi n}{q}\right $$ where $0<p<q$ .
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"Gauss' digamma theorem" is owned by rm50.
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Cross-references: Gauss, theorem
This is version 3 of Gauss' digamma theorem, born on 2006-11-11, modified 2006-11-12.
Object id is 8542, canonical name is GaussDigammaTheorem.
Accessed 1028 times total.
Classification:
| AMS MSC: | 33B15 (Special functions :: Elementary classical functions :: Gamma, beta and polygamma functions) | | | 30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory) |
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Pending Errata and Addenda
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