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A Computational Procedure for Incomplete Gamma Functions
ACM Transactions on Mathematical Software, 1979We develop a computational procedure, based on Taylor's series and continued fractions, for evaluating Tncomi's incomplete gamma functmn 7*(a, x) = (x-"/F(a))S~ e-~t'-ldt and the complementary incomplete gamma function F(a, x) = $7 e-tt "-1 dt, suitably normalized, m the region x >_. 0, -oo < a < oo.
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The Gamma Function and the Incomplete Gamma Functions
2017The gamma function is defined for \(s \in \mathbb{C}\) by $$\displaystyle{ \varGamma \left (s\right ) =\int _{ 0}^{\infty }t^{s-1}e^{-t}dt }$$
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On Defining The Incomplete Gamma Function
Integral Transforms and Special Functions, 2003The incomplete Gamma function $\gamma \lpar \alpha\comma \; x_+\rpar $ is defined as locally summable function on the real line for $\alpha \gt 0$ by $$\gamma \lpar \alpha\comma \; x_+\rpar =\vint_0^{x_+} u^{\alpha -1} e^{-u}\, \hbox{d}u\comma$$ the integral diverging for $\alpha\leq 0$ .
Fisher, Brian +2 more
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Inequalities for the Incomplete Gamma and Related Functions
Mathematical Inequalities & Applications, 1999The authors offer lower and upper estimates for \(\int_{0}^{x}e^{t^{p}}dt\) and for the similar function with \(-t^{p}\) in the exponent, furthermore examples showing that these are not comparable to those found by \textit{H. Alzer} (Math.Comp. 66, 771-778 (1997; Zbl 0865.33002).
Qi, Feng, Guo, Sen-Lin
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Some bounds for incomplete gamma function
Rendiconti del Circolo Matematico di Palermo, 1994Take for the incomplete gamma function with parameter \(a\) the alternating series [see, e.g., \textit{M. Abramowitz} and \textit{I. A. Stegun}, Handbook of mathematical functions with formulas, graphs, and mathematical tables. (1964; Zbl 0171.385), p.
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Inequalities and Bounds for the Incomplete Gamma Function
Results in Mathematics, 2012zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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CLT, LDP and incomplete gamma functions
2019Given a series of identically distributed random variables X?, X?, ..., X? with mean ? and variance ? , the CLT(Central Limit Theorem) can provide a good approximation of the probability ?(|X?? ? ?|) > x for x which is not too big compared to ???n. Although the CLT always works perfectly around the ?center?
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Composition and functions of bacterial membrane vesicles
Nature Reviews Microbiology, 2023Masanori Toyofuku +2 more
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Gene regulation by long non-coding RNAs and its biological functions
Nature Reviews Molecular Cell Biology, 2020Luisa Statello +2 more
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