A System of Inequalities for the Incomplete Gamma Function and the Normal Integral
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Abstract:
In this paper a new set of inequalities and bounds for the incomplete gamma function are obtained. These inequalities and bounds are based on continued fraction expansions of the incomplete gamma function (Sections 2 and 3). Comparisons between the two sets of inequalities and some other known inequalities are made (Section 4). Bounds are also obtained for the Mills' ratio for the normal integral (Section 5) and an analogue of Mills' ratio (Section 6) for the gamma distribution. Some other applications of these bounds to distribution theory problems arising in multiple decision theory are described (Section 6).Keywords:
Incomplete gamma function
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The multiple gamma function $\Gamma_n$, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the $\Gamma_n$ function and their applications to summation of series and infinite products.
Incomplete gamma function
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Liouville’s theorem proves that certain integrals cannot be evaluated with elementary functions. It demonstrates why the gamma, exponential and Gaussian integrals lack antiderivatives. However, by applying the “h” factorization method, the author presents an analytical solution to the antiderivative of the gamma integral. This solution applies to all integrals that can be transformed into a gamma integral, including the exponential and Gaussian integrals. The author further provides a thorough discussion of the algebraic properties of the “h” function. The major contribution to statistical science is that “h” can serve as the most fundamental function which unifies many cumulative distribution functions, such as the gamma function, the exponential integral function, the error function, the beta function, the hypergeometric function, and the Marcum Q-function, and the truncated normal distribution. The closed-form expression of the moment-generating function for the truncated normal distribution can be also derived as an “h” function.
Exponential integral
Incomplete gamma function
Gaussian integral
Mittag-Leffler function
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In this paper we study the integral of type
\[_{\delta,a}\Gamma_{\rho,b}(x) =\Gamma(\delta,a;\rho,b)(x)=\int_{0}^{\infty}t^{x-1}e^{-\frac{t^{\delta}}{a}-\frac{t^{-\rho}}{b}}dt.\] Different authors called this integral by different names like ultra gamma function, generalized gamma function, Kratzel integral, inverse Gaussian integral, reaction-rate probability integral, Bessel integral etc. We prove several identities and recurrence relation of above said integral, we called this integral as Four Parameter Gamma Function. Also we evaluate relation between Four Parameter Gamma Function, p-k Gamma Function and Classical Gamma Function. With some conditions we can evaluate Four Parameter Gamma Function in term of Hypergeometric function.
Incomplete gamma function
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We apply our simultaneous contour integral method to an infinite sum in Prudnikov et al. and use it to derive the infinite sum of the Incomplete gamma function in terms of the Hurwitz zeta function. We then evaluate this formula to derive new series in terms of special functions and fundamental constants. All the results in this work are new.
Incomplete gamma function
Digamma function
Polylogarithm
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Incomplete gamma function
Exponential integral
Series expansion
Exponential formula
Elementary function
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Multiple integral
Incomplete gamma function
Representation
Triple product
Exponential integral
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In this paper we study the integral of type \[_{\delta,a}\Gamma_{\rho,b}(x) =\Gamma(\delta,a;\rho,b)(x)=\int_{0}^{\infty}t^{x-1}e^{-\frac{t^{\delta}}{a}-\frac{t^{-\rho}}{b}}dt.\] Different authors called this integral by different names like ultra gamma function, generalized gamma function, Kratzel integral, inverse Gaussian integral, reaction-rate probability integral, Bessel integral etc. We prove several identities and recurrence relation of above said integral, we called this integral as Four Parameter Gamma Function. Also we evaluate relation between Four Parameter Gamma Function, p-k Gamma Function and Classical Gamma Function. With some conditions we can evaluate Four Parameter Gamma Function in term of Hypergeometric function.
Incomplete gamma function
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As a generalization of the Gamma function defined for a single complex variable, a new special function called a generalized Gamma function, defined for two complex variables and a positive integer, is introduced, and several important analytical properties are investigated in detail, which include regularity, asymptotic expansions and analytic continuations. Furthermore, as a function closely related to a generalized Gamma function, a generalized incomplete Gamma function, which is a generalization of the incomplete Gamma function, is also introduced, and some fundamental properties are investigated briefly.
Incomplete gamma function
Generalized function
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We consider the asymptotic behavior of the incomplete gamma functions $gamma (-a,-z)$ and $Gamma (-a,-z)$ as $atoinfty$. Uniform expansions are needed to describe the transition area $z sim a$, in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform expansion by applying techniques used for the existing uniform expansions for $gamma (a,z)$ and $Gamma (a,z)$. The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given in a final section.
Incomplete gamma function
Asymptotic expansion
Series expansion
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Incomplete gamma function
Digamma function
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Beta function (physics)
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