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On Defining The Incomplete Gamma Function
Integral Transforms and Special Functions, 2003 The 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 $\alpha\leq 0$ .
Adem Kılıçman
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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.
Edward Neuman
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Efficient and Accurate Algorithms for the Computation and Inversion of the Incomplete Gamma Function Ratios [PDF]
SIAM Journal of Scientific Computing, 2012 Algorithms for the numerical evaluation of the incomplete gamma function ratios $P(a,x)=\gamma(a,x)/\Gamma(a)$ and $Q(a,x)=\Gamma(a,x)/\Gamma(a)$ are described for positive values of $a$ and $x$.
Javier Segura, Amparo Gil, Nico M Temme
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Functional inequalities for the incomplete gamma function
Journal of Mathematical Analysis and Applications, 2012 Let \(f_a(x)= \Gamma(a,x)/\Gamma(a,0)\), where \(\Gamma(a,x)\) denotes the incomplete gamma function \((a,x> 0)\). The authors prove various new functional inequalities for \(f_a(x)\). For example, they study the double inequality \[ f_a(S_p(x_1,\dots, x_n))\leq f_a(x_1)\cdots f_a(x_n)\leq f_a(S_q(x_1,\dots, x_n)), \] where \(S_t\) is the power sum of ...
Horst Alzer, Árpád Baricz
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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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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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The incomplete gamma functions
The Mathematical Gazette, 2016Summary: Recall the integral definition of the gamma function: \(\Gamma(a)=\int_0^{\infty}t^{a-1}e^{-t}dt\) for \(a>0\). By splitting this integral at a point \(x\geq 0\), we obtain the two \textit{incomplete gamma functions}: \[ \gamma(a,x)=\int_0^xt^{a-1}e^{-t}dt,\eqno{(1)} \] \[ \Gamma(a,x)=\int_x^{\infty}t^{a-1}e^{-t}dt\eqno{(2)} \] \(\Gamma(a,x)\)
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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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