# Fungrim entry: b01280

$\frac{\log^{k}\!\left(1 + x\right)}{k !} = \sum_{n=k}^{\infty} {\left(-1\right)}^{n - k} \left[{n \atop k}\right] \frac{{x}^{n}}{n !}$
Assumptions:$k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 1$
TeX:
\frac{\log^{k}\!\left(1 + x\right)}{k !} = \sum_{n=k}^{\infty} {\left(-1\right)}^{n - k} \left[{n \atop k}\right] \frac{{x}^{n}}{n !}

k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 1
Definitions:
Fungrim symbol Notation Short description
Pow${a}^{b}$ Power
Log$\log(z)$ Natural logarithm
Factorial$n !$ Factorial
Sum$\sum_{n} f(n)$ Sum
StirlingCycle$\left[{n \atop k}\right]$ Unsigned Stirling number of the first kind
Infinity$\infty$ Positive infinity
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
Abs$\left|z\right|$ Absolute value
Source code for this entry:
Entry(ID("b01280"),
Formula(Equal(Div(Pow(Log(Add(1, x)), k), Factorial(k)), Sum(Mul(Mul(Pow(-1, Sub(n, k)), StirlingCycle(n, k)), Div(Pow(x, n), Factorial(n))), For(n, k, Infinity)))),
Variables(x, k),
Assumptions(And(Element(k, ZZGreaterEqual(0)), Element(x, CC), Less(Abs(x), 1))))

## 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