# Fungrim entry: cf5355

$\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - \sum_{n=1}^{N - 1} \frac{B_{2 n}}{2 n {z}^{2 n}} + R'_{N}(z)$
Assumptions:$z \in \mathbb{C} \setminus \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 0}$
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
DigammaFunction$\psi\!\left(z\right)$ Digamma function
Log$\log(z)$ Natural logarithm
Sum$\sum_{n} f(n)$ Sum
BernoulliB$B_{n}$ Bernoulli number
Pow${a}^{b}$ Power
Derivative$\frac{d}{d z}\, f\!\left(z\right)$ Derivative
StirlingSeriesRemainder$R_{n}\!\left(z\right)$ Remainder term in the Stirling series for the logarithmic gamma function
CC$\mathbb{C}$ Complex numbers
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Infinity$\infty$ Positive infinity
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ 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)))))

## Topics using this entry

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