The Gamma Function and the Incomplete Gamma Functions

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.

Volume 5: 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C ◽

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

10.1115/detc2007-35843 ◽

2007 ◽

Cited By ~ 4

Author(s):

Tom T. Hartley ◽

Carl F. Lorenzo

Keyword(s):

System Theory ◽

Fractional Order ◽

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

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.2833480 ◽

2008 ◽

Vol 3 (2) ◽

Cited By ~ 9

Author(s):

Tom T. Hartley ◽

Carl F. Lorenzo

Keyword(s):

Fractional Order ◽

Gamma Function ◽

Order Differential Equation ◽

Order System ◽

Fractional Order Differential Equation ◽

Laplace Domain ◽

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.

A Set of Algorithms for the Incomplete Gamma Functions

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800003417 ◽

1994 ◽

Vol 8 (2) ◽

pp. 291-307 ◽

Cited By ~ 8

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.

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

Analysis and Applications ◽

10.1142/s0219530517500099 ◽

2018 ◽

Vol 16 (03) ◽

pp. 435-448 ◽

Cited By ~ 1

Author(s):

Blanca Bujanda ◽

José L. López ◽

Pedro J. Pagola

Keyword(s):

Gamma Function ◽

Error Bounds ◽

Rational Functions ◽

Elementary Functions ◽

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.

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

Journal of Inequalities and Applications ◽

10.1186/s13660-020-02538-y ◽

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 ◽

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.