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# Fungrim entry: 411f3b

$\gamma_{n}\!\left(a\right) = \lim_{N \to \infty} \left[\left(\sum_{k=0}^{N} \frac{\log^{n}\!\left(k + a\right)}{k + a}\right) - \frac{\log^{n + 1}\!\left(N + a\right)}{n + 1}\right]$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}$
TeX:
\gamma_{n}\!\left(a\right) = \lim_{N \to \infty} \left[\left(\sum_{k=0}^{N} \frac{\log^{n}\!\left(k + a\right)}{k + a}\right) - \frac{\log^{n + 1}\!\left(N + a\right)}{n + 1}\right]

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \notin \{0, -1, \ldots\}
Definitions:
Fungrim symbol Notation Short description
StieltjesGamma$\gamma_{n}\!\left(a\right)$ Stieltjes constant
SequenceLimit$\lim_{n \to a} f(n)$ Limiting value of sequence
Sum$\sum_{n} f(n)$ Sum
Pow${a}^{b}$ Power
Log$\log(z)$ Natural logarithm
Infinity$\infty$ Positive infinity
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
Source code for this entry:
Entry(ID("411f3b"),
Formula(Equal(StieltjesGamma(n, a), SequenceLimit(Brackets(Sub(Parentheses(Sum(Div(Pow(Log(Add(k, a)), n), Add(k, a)), For(k, 0, N))), Div(Pow(Log(Add(N, a)), Add(n, 1)), Add(n, 1)))), For(N, Infinity)))),
Variables(n, a),
Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), NotElement(a, ZZLessEqual(0)))))

## Topics using this entry

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