Fungrim home page

Fungrim entry: b01280

logk ⁣(1+x)k!=n=k(1)nk[nk]xnn!\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:kZ0  and  xC  and  x<1k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 1
\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
Fungrim symbol Notation Short description
Powab{a}^{b} Power
Loglog(z)\log(z) Natural logarithm
Factorialn!n ! Factorial
Sumnf(n)\sum_{n} f(n) Sum
StirlingCycle[nk]\left[{n \atop k}\right] Unsigned Stirling number of the first kind
Infinity\infty Positive infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
Source code for this entry:
    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.

2020-08-27 09:56:25.682319 UTC