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Fungrim entry: 3fe553

ψ ⁣(pq)=γlog ⁣(2q)π2cot ⁣(πpq)+2k=1(q1)/2cos ⁣(2πkpq)log ⁣(sin ⁣(πkq))\psi\!\left(\frac{p}{q}\right) = -\gamma - \log\!\left(2 q\right) - \frac{\pi}{2} \cot\!\left(\frac{\pi p}{q}\right) + 2 \sum_{k=1}^{\left\lfloor \left( q - 1 \right) / 2 \right\rfloor} \cos\!\left(\frac{2 \pi k p}{q}\right) \log\!\left(\sin\!\left(\frac{\pi k}{q}\right)\right)
Assumptions:qZ2  and  p{1,2,,q1}q \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; p \in \{1, 2, \ldots, q - 1\}
\psi\!\left(\frac{p}{q}\right) = -\gamma - \log\!\left(2 q\right) - \frac{\pi}{2} \cot\!\left(\frac{\pi p}{q}\right) + 2 \sum_{k=1}^{\left\lfloor \left( q - 1 \right) / 2 \right\rfloor} \cos\!\left(\frac{2 \pi k p}{q}\right) \log\!\left(\sin\!\left(\frac{\pi k}{q}\right)\right)

q \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; p \in \{1, 2, \ldots, q - 1\}
Fungrim symbol Notation Short description
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
ConstGammaγ\gamma The constant gamma (0.577...)
Loglog(z)\log(z) Natural logarithm
Piπ\pi The constant pi (3.14...)
Sumnf(n)\sum_{n} f(n) Sum
Coscos(z)\cos(z) Cosine
Sinsin(z)\sin(z) Sine
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
Source code for this entry:
    Formula(Equal(DigammaFunction(Div(p, q)), Add(Sub(Sub(Neg(ConstGamma), Log(Mul(2, q))), Mul(Div(Pi, 2), Cot(Div(Mul(Pi, p), q)))), Mul(2, Sum(Mul(Cos(Div(Mul(Mul(Mul(2, Pi), k), p), q)), Log(Sin(Div(Mul(Pi, k), q)))), For(k, 1, Floor(Div(Sub(q, 1), 2)))))))),
    Variables(p, q),
    Assumptions(And(Element(q, ZZGreaterEqual(2)), Element(p, Range(1, Sub(q, 1))))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC