scholarly journals Convergent expansions of the incomplete gamma functions in terms of elementary functions

2018 ◽  
Vol 16 (03) ◽  
pp. 435-448 ◽  
Author(s):  
Blanca Bujanda ◽  
José L. López ◽  
Pedro J. Pagola
Keyword(s):  
Gamma Function ◽  
Error Bounds ◽  
Rational Functions ◽  
Incomplete Gamma Function ◽  
Elementary Functions ◽  
Gamma Functions ◽  
Incomplete Gamma Functions ◽  
Uniformly Convergent

We consider the incomplete gamma function [Formula: see text] for [Formula: see text] and [Formula: see text]. We derive several convergent expansions of [Formula: see text] in terms of exponentials and rational functions of [Formula: see text] that hold uniformly in [Formula: see text] with [Formula: see text] bounded from below. These expansions, multiplied by [Formula: see text], are expansions of [Formula: see text] uniformly convergent in [Formula: see text] with [Formula: see text] bounded from above. The expansions are accompanied by realistic error bounds.

The incomplete gamma functions

2016 ◽  
Vol 100 (548) ◽  
pp. 298-306 ◽  
Author(s):  
G. J. O. Jameson
Keyword(s):  
Gamma Function ◽  
Incomplete Gamma Function ◽  
Gamma Functions ◽  
Incomplete Gamma Functions ◽  
Definition Of ◽  
Image Position ◽  
Selection Of

Recall the integral definition of the gamma function: for a > 0. By splitting this integral at a point x ⩾ 0, we obtain the two incomplete gamma functions:(1)(2)Γ(a, x)is sometimes called the complementary incomplete gamma function. These functions were first investigated by Prym in 1877, and Γ(a, x) has also been called Prym's function. Not many books give these functions much space. Massive compilations of results about them can be seen stated without proof in [1, chapter 9] and [2, chapter 8]. Here we offer a small selection of these results, with proofs and some discussion of context. We hope to convince some readers that the functions are interesting enough to merit attention in their own right.

Application of Incomplete Gamma Functions to the Initialization of Fractional-Order Systems

2007 ◽  
Author(s):  
Tom T. Hartley ◽  
Carl F. Lorenzo
Keyword(s):  
System Theory ◽  
Fractional Order ◽  
Gamma Function ◽  
Incomplete Gamma Function ◽  
Fractional Order System ◽  
Order System ◽  
Fractional Order Systems ◽  
Gamma Functions ◽  
Incomplete Gamma Functions

This paper reviews some properties of the gamma function, particularly the incomplete gamma function and its complement, as a function of the Laplace variable s. The utility of these functions in the solution of initialization problems in fractional-order system theory is demonstrated.

Application of Incomplete Gamma Functions to the Initialization of Fractional-Order Systems

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Tom T. Hartley ◽  
Carl F. Lorenzo
Keyword(s):  
Fractional Order ◽  
Gamma Function ◽  
Order Differential Equation ◽  
Incomplete Gamma Function ◽  
Order System ◽  
Fractional Order Differential Equation ◽  
Laplace Domain ◽  
Gamma Functions ◽  
Incomplete Gamma Functions ◽  
The Time Domain

This paper reviews some properties of the gamma function, particularly the incomplete gamma function and its complement, as a function of the Laplace variable s. The utility of these functions in the solution of initialization problems in fractional-order system theory is demonstrated. Several specific differential equations are presented, and their initialization responses are found for a variety of initializations. Both the time-domain and Laplace-domain solutions are obtained and compared. The complementary incomplete gamma function is shown to be essential in finding the Laplace-domain solution of a fractional-order differential equation.

scholarly journals The resurgence properties of the incomplete gamma function, I

2016 ◽  
Vol 14 (05) ◽  
pp. 631-677 ◽  
Author(s):  
Gergő Nemes
Keyword(s):  
Gamma Function ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Incomplete Gamma Function ◽  
Smooth Transition

In this paper, we derive new representations for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by C. J. Howls [Hyperasymptotics for integrals with finite endpoints, Proc. Roy. Soc. London Ser. A 439 (1992) 373–396]. Using these representations, we obtain a number of properties of the asymptotic expansions of the incomplete gamma function with large arguments, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.

scholarly journals A Set of Algorithms for the Incomplete Gamma Functions

1994 ◽  
Vol 8 (2) ◽  
pp. 291-307 ◽  
Author(s):  
N. M. Temme
Keyword(s):  
Gamma Function ◽  
Numerical Evaluation ◽  
Error Function ◽  
Applied Probability ◽  
Gamma Functions ◽  
Auxiliary Functions ◽  
Incomplete Gamma Functions

This paper gives fast and reliable algorithms for the numerical evaluation of the incomplete gamma functions and for auxiliary functions, such as functions related with the gamma function and error function. All these functions are of basic importance in applied probability problems.

scholarly journals Extended incomplete version of hypergeometric functions

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 653-662
Author(s):  
Mehmet Özarslan ◽  
Ceren Ustaoğlu
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Integral Representations ◽  
Elementary Functions ◽  
Gamma Functions ◽  
Mellin Transforms ◽  
Incomplete Gamma Functions ◽  
Derivative Formulas ◽  
Transformation Formulas

Recently, the incomplete Pochhammer ratios are defined in terms of incomplete beta and gamma functions [10]. In this paper, we introduce the extended incomplete version of Pochhammer symbols in terms of the generalized incomplete gamma functions. With the help of this extended incomplete version of Pochhammer symbols we introduce the extended incomplete version of Gauss hypergeometric and Appell?s functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, Mellin transforms and log convex properties. Furthermore, we investigate incomplete fractional derivatives for extended incomplete version of some elementary functions.

ON THE INCOMPLETE GAMMA FUNCTION AND ITS NEUTRIX CONVOLUTION FOR NEGATIVE INTEGERS

2019 ◽  
Vol 10 (1) ◽  
pp. 30-51
Author(s):  
Mongkolsery Lin ◽  
◽  
Brian Fisher ◽  
Somsak Orankitjaroen ◽  
◽  
...  
Keyword(s):  
Gamma Function ◽  
Incomplete Gamma Function

scholarly journals New fractional inequalities of Hermite–Hadamard type involving the incomplete gamma functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Dumitru Baleanu ◽  
Artion Kashuri ◽  
Faraidun Hamasalh ◽  
...  
Keyword(s):  
Convex Functions ◽  
Special Functions ◽  
Integral Inequalities ◽  
Fractional Operators ◽  
Gamma Functions ◽  
Incomplete Gamma Functions ◽  
Type Integral

AbstractA specific type of convex functions is discussed. By examining this, we investigate new Hermite–Hadamard type integral inequalities for the Riemann–Liouville fractional operators involving the generalized incomplete gamma functions. Finally, we expose some examples of special functions to support the usefulness and effectiveness of our results.

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