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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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An algorithm for the evaluation of the incomplete gamma function
Advances in Computational Mathematics, 2018For real numbers \(m>0\) and \(x>0\) denote by \(\gamma(m,x)\) the incomplete gamma function given by \[ \gamma(m,x)=\int_0^x t^{m-1} e^{-t} dt \] and let \(P(m,x)\) be the incomplete gamma function normalized by the gamma function, that is, \[ P(m,x)=\frac{\gamma(m,x)}{\Gamma(m)}. \] The authors provide formulas for evaluating \(P(m,x)\).
Philip Greengard, Vladimir Rokhlin
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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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1989
This table contains values of \(F(x;\alpha ) = \int_0^x {{1 \over {\Gamma (\alpha )}}{y^{\alpha -1}}{e^{ -y}}dy.} \)
Stephen Kokoska, Christopher Nevison
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This table contains values of \(F(x;\alpha ) = \int_0^x {{1 \over {\Gamma (\alpha )}}{y^{\alpha -1}}{e^{ -y}}dy.} \)
Stephen Kokoska, Christopher Nevison
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Grünbaum-Type Inequalities for Gamma and Incomplete Gamma Functions
Results in Mathematics, 2018zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Alzer, Horst, Kwong, Man Kam
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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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Products of incomplete gamma functions
Analysis, 2015AbstractMany properties of gamma functions are known. In this paper, we extend similar properties to incomplete gamma functions.
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Extension of Incomplete Gamma, Beta and Hypergeometric Functions
Progress in Fractional Differentiation and Applications, 2019Recently, some extensions of the generalised gamma, beta, Gauss hypergeometric and confluent hypergeometric functions have been introduced. In this paper, we introduce generalisations of incomplete gamma, beta, Gauss, confluent and Appell's hypergeometric functions.
Özarslan, Mehmet Ali, Ustaoğlu, Ceren
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The Incomplete Gamma Function Expressed as a Sum of Macdonald Functions
Results in Mathematics, 2001The author uses the expansion of the incomplete gamma function defined by \[ \gamma(\nu,z)=\sqrt{{2\over\pi}}\int_0^z s^{\nu-1/2} K_{1/2}(s) ds,\quad \Re \nu>0, \] in terms of the modified Bessel functions of the second kind \(K_{n+1/2}(z)\). This expansion allows to evaluate different types of integrals of interest in atomic physics.
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A note on the recursive calculation of incomplete gamma functions
ACM Transactions on Mathematical Software, 1999It is known that the recurrence relation for incomplete gamma functions {γ( a + n , x )}, 0 ≤ a < 1, n = 0, 1, 2 ..., when x is positive, is unstable—more so the larger x
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