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Fungrim entry: cf5355

ψ ⁣(z)=log(z)12zn=1N1B2n2nz2n+RN(z)\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - \sum_{n=1}^{N - 1} \frac{B_{2 n}}{2 n {z}^{2 n}} + R'_{N}(z)
Assumptions:zC(,0]  and  NZ0z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 0}
TeX:
\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - \sum_{n=1}^{N - 1} \frac{B_{2 n}}{2 n {z}^{2 n}} + R'_{N}(z)

z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
Loglog(z)\log(z) Natural logarithm
Sumnf(n)\sum_{n} f(n) Sum
BernoulliBBnB_{n} Bernoulli number
Powab{a}^{b} Power
Derivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Derivative
StirlingSeriesRemainderRn ⁣(z)R_{n}\!\left(z\right) Remainder term in the Stirling series for the logarithmic gamma function
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("cf5355"),
    Formula(Equal(DigammaFunction(z), Add(Sub(Sub(Log(z), Div(1, Mul(2, z))), Sum(Div(BernoulliB(Mul(2, n)), Mul(Mul(2, n), Pow(z, Mul(2, n)))), For(n, 1, Sub(N, 1)))), Derivative(StirlingSeriesRemainder(N, z), For(z, z))))),
    Variables(z, N),
    Assumptions(And(Element(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))), Element(N, ZZGreaterEqual(0)))))

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

2020-08-27 09:56:25.682319 UTC