# Fungrim entry: cb5071

$\left|\frac{1}{n !} \left[ \frac{d^{n}}{{d x}^{n}} \frac{1}{\Gamma(x)} \right]_{x = 0}\right| \le \frac{2}{\sqrt{n !}}$
Assumptions:$n \in \mathbb{Z}_{\ge 0}$
References:
• L. Fekih-Ahmed, On the Power Series Expansion of the Reciprocal Gamma Function, https://arxiv.org/abs/1407.5983 (simplified version of (1.5))
TeX:
\left|\frac{1}{n !} \left[ \frac{d^{n}}{{d x}^{n}} \frac{1}{\Gamma(x)} \right]_{x = 0}\right| \le \frac{2}{\sqrt{n !}}

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
Factorial$n !$ Factorial
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
Gamma$\Gamma(z)$ Gamma function
Sqrt$\sqrt{z}$ Principal square root
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("cb5071"),
Formula(LessEqual(Abs(Mul(Div(1, Factorial(n)), ComplexDerivative(Div(1, Gamma(x)), For(x, 0, n)))), Div(2, Sqrt(Factorial(n))))),
Variables(n),
Assumptions(And(Element(n, ZZGreaterEqual(0)))),
References("L. Fekih-Ahmed, On the Power Series Expansion of the Reciprocal Gamma Function, https://arxiv.org/abs/1407.5983 (simplified version of (1.5))"))

## 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