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About 783 papers for “Incomplete gamma function”

Tables of the incomplete [Gamma]-function,

Pearson, Karl, 1857-1936., Great Britain. Department of Scientific and Industrial Research., University College, London. Department of Applied Statistics. · The Journal of the Abraham Lincoln Association

Cited by 0Open access

Algorithm 435: modified incomplete gamma function [S14]

Wayne Fullerton · Communications of the ACM · 1972

article Free AccessAlgorithm 435: modified incomplete gamma function [S14] Author: Wayne Fullerton Univ. of Michigan, Ann Arbor Univ. of Michigan, Ann ArborView Profile Authors Info & Claims Communications of the ACMVolume 15Issue 1101 November 1972pp 993–995https://doi.org/10.1145/355606.361891Published:01 November 1972Publication History 4citation331DownloadsMetricsTotal Citations4Total Downloads331Last 12 Months19Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Cited by 4Open access

Landmark Computation of the Incomplete Gamma Function

Ilir F. Progri · Journal of Geolocation Geo-information and Geo-intelligence · 2022

Use in connection with any form of information storage and retrieval, electronic adaptation, computer software or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use of the publication of trade names, trademarks, service marks, or similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether they are subject to proprietary rights.The landmark i computation of the regularized ii incomplete gamma function (RIGF) provides a novel, innovative, new, and significantly improved understanding of the regularization of the IGF by means of new integral expressions and describes (or gives a very detailed account) on the scale of validity of the range of several already known closed form expressions of the computation of the RIGF.The lack of the convergence range on the computation of the IGF is the root cause of the computational issues that were described in the quantitative evaluation of the generalized Bessel function distribution of the first kind (GBFD1K).The landmark computation of the RIGF depicts that the landmark RIGF is in fact an integration of the exponential function with a more complicated exponent; therefore, the convergence of the power series expansion of the exponential function integration for negative exponents has directly impacted the computation of the RIGF.In order to compare (or collate and contrast) the computation of the RIGF with the available literature approaches we used three methods: analytical, graphical, and numerical (or computationally numerical via MATLAB simulations).As a result of this work, we provided a well-defined range of convergence for the closed form expressions of the computation of the RIGF in the literature and we have significantly improved the understanding of the computation of the RIGF.

Cited by 7Open access

Computation of the Incomplete Gamma Function Ratios

A. R. DiDonato, R. K. Hageman · 1976

Abstract : A method, made up of several algorithms, is given for computing the incomplete gamma function ratio, P(a, x) and its complement Q(a, x) for all real arguments a 0, x or = 0. The difficult case when a and x are large is treated by a modified version of Takenaga's method. The resulting computer program is efficient, yields both P and Q correctly to within 1 unit in the twelfth significant digit, or, at the user's option, to within one unit in the sixth or third significant digit. A FORTRAN listing of the program is included.

Cited by 5

Some inequalities related to incomplete gamma function

Gholam Reza Mohtashami Borzadaran, Mohammad Reza Miri · Journal of Interdisciplinary Mathematics · 2007

Abstract In this note, inequalities related to the versions of the incomplete gamma function that is introduced in a review paper by Gautschi (1998) are derived in view of the arguments in Merkle (1993).

Cited by 1

expint: Exponential Integral and Incomplete Gamma Function

Vincent Goulet, R Core Team, Gerard Jungman et al. · 2016

The exponential integrals E_1(x), E_2(x), E_n(x) and Ei(x), and the incomplete gamma function G(a, x) defined for negative values of its first argument. The package also gives easy access to the underlying C routines through an API; see the package vignette for details. A test package included in sub-directory example_API provides an implementation. C routines derived from the GNU Scientific Library &lt;<a href="https://www.gnu.org/software/gsl/" target="_top">https://www.gnu.org/software/gsl/</a>&gt;.

Cited by 8Open access

Inverse incomplete gamma function and its application

Mischa Döhler, Marylin Arndt · Electronics Letters · 2006

Communication systems with multiple antennas operating over non-ergodic fading channels are gauged by means of the outage probability against a given communication rate, the solution to which is often expressed in terms of the incomplete gamma function. Practical situations, however, often require the inverse process, i.e. the determination of an optimum rate against a desired outage probability. The contribution of this report is the application of an asymptotic expression that enables the inverse function to be expressed in closed form.

Cited by 34

Monotonicity of the incomplete gamma function with applications

Zhen-Hang Yang, Wen Zhang, Yu‐Ming Chu · Journal of Inequalities and Applications · 2016

In the article, we discuss the monotonicity properties of the function $x\rightarrow (1-e^{-ax^{p}} )^{1/p}/\int_{0}^{x}e^{-t^{p}}\,dt$ for $a, p>0$ with $p\neq1$ on $(0, \infty)$ and prove that the double inequality $\Gamma(1+1/p) (1-e^{-a x^{p}} )^{1/p} <\int_{0}^{x}e^{-t^{p}}\,dt<\Gamma(1+1/p) (1-e^{-b x^{p}} )^{1/p}$ holds for all $x>0$ if and only if $a\leq\min\{1, \Gamma^{-p}(1+1/p)\}$ and $b\geq\max\{1, \Gamma^{-p}(1+1/p)\}$ .

Cited by 10Open access

On some inequalities for the incomplete gamma function

Horst Alzer · Mathematics of Computation · 1997

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p not-equals 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p\ne 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a positive real number. We determine all real numbers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha equals alpha left-parenthesis p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha = \alpha (p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta equals beta left-parenthesis p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mi>β</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta =\beta (p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the inequalities <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 1 minus e Superscript minus beta x Super Superscript p Superscript Baseline right-bracket Superscript 1 slash p Baseline greater-than StartFraction 1 Over normal upper Gamma left-parenthesis 1 plus 1 slash p right-parenthesis EndFraction integral Subscript 0 Superscript x Baseline e Superscript minus t Super Superscript p Baseline d t greater-than left-bracket 1 minus e Superscript minus alpha x Super Superscript p Superscript Baseline right-bracket Superscript 1 slash p"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mi>β</mml:mi> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:msup> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo>&gt;</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfrac> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mn>0</mml:mn> <mml:mi>x</mml:mi> </mml:msubsup> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:msup> <mml:mspace width="thinmathspace"/> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mi>α</mml:mi> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:msup> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">[1-e^{-\beta x^p}]^{1/p}&gt; \frac 1{\Gamma (1+1/p)} \int ^x_0 e^{-t^p} \,dt &gt;[1-e^{-\alpha x^p}]^{1/p}</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> are valid for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">x&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. And, we determine all real numbers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding="application/x-tex">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus log left-parenthesis 1 minus e Superscript minus a x Baseline right-parenthesis less-than-or-equal-to integral Subscript x Superscript normal infinity Baseline StartFraction e Superscript negative t Baseline Over t EndFraction d t less-than-or-equal-to minus log left-parenthesis 1 minus e Superscript minus b x Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mi>a</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≤</mml:mo> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mi>x</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msubsup> <mml:mfrac> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:mi>t</mml:mi> </mml:mfrac> <mml:mspace width="thinmathspace"/> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>≤</mml:mo> <mml:mo>−</mml:mo> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mi>b</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">-\log (1-e^{-ax})\le \int ^\infty _x \frac {e^{-t}}t\,dt\le -\log (1-e^{-bx})</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> hold for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">x&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Cited by 390Open access

Derangements and the $p$-adic incomplete gamma function

Andrew O’Desky, David Harry Richman · arXiv (Cornell University) · 2020

We introduce a $p$-adic analogue of the incomplete gamma function. We also introduce quantities ($m$-values) associated to a function on natural numbers and prove a new characterization of $p$-adic continuity for functions with $p$-integral $m$-values. Combinatorial interpretations for the integral values of the incomplete gamma function and functions with $m$-values zero or one are obtained, which show that these functions count derangements in generalized symmetric groups and permutations with restricted cycle lengths.

Cited by 0Open access

Monotonicity and inequalities involving the incomplete gamma function

Zhen-Hang Yang, Wen Zhang, Yu‐Ming Chu · Journal of Inequalities and Applications · 2016

In the article, we deal with the monotonicity of the function $x\rightarrow[ (x^{p}+a )^{1/p}-x]/I_{p}(x)$ on the interval $(0, \infty)$ for $p>1$ and $a>0$ , and present the necessary and sufficient condition such that the double inequality $[ (x^{p}+a )^{1/p}-x]/a< I_{p}(x)<[ (x^{p}+b )^{1/p}-x]/b$ for all $x>0$ and $p>1$ , where $I_{p}(x)=e^{x^{p}}\int_{x}^{\infty}e^{-t^{p}}\,dt$ is the incomplete gamma function.

Cited by 9Open access