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Fungrim entry: cb5071

1n![dndxn1Γ(x)]x=02n!\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:nZ0n \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
Absz\left|z\right| Absolute value
Factorialn!n ! Factorial
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
GammaΓ(z)\Gamma(z) Gamma function
Sqrtz\sqrt{z} Principal square root
ZZGreaterEqualZn\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))"))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-11-19 15:10:20.037976 UTC