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 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.

Ramanujan's Continued Fraction and the Bauer-Muir Transformation

1961 ◽  
Vol 57 (1) ◽  
pp. 76-79 ◽  
Author(s):  
V. N. Singh
Keyword(s):  
Positive Integer ◽  
Gamma Function ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Gamma Functions ◽  
Image Position ◽  
The Right ◽  
Basic Analogue ◽  
Ramanujan's Continued Fraction

Ramanujan's Continued Fraction may be stated as follows: Let where there are eight gamma functions in each product and the ambiguous signs are so chosen that the argument of each gamma function contains one of the specified number of minus signs. Then where the products and the sums on the right range over the numbers α, β, γ, δ, ε: provided that one of the numbers β, γ, δ, ε is equal to ± ±n, where n is a positive integer. In 1935, Watson (3) proved the theorem by induction and also gave a basic analogue. In this paper we give a new proof of Ramanujan's Continued Fraction by using the transformation of Bauer and Muir in the theory of continued fractions (Perron (1), §7;(2), §2).

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.

Ramanujan's Continued Fraction

1935 ◽  
Vol 31 (1) ◽  
pp. 7-17 ◽  
Author(s):  
G. N. Watson
Keyword(s):  
Positive Integer ◽  
Gamma Function ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Gamma Functions ◽  
Image Position ◽  
The Right ◽  
Ramanujan's Continued Fraction

The best of the theorems on continued fractions, to be found in Ramanujan's manuscript note-book may be stated as follows:where there are eight gamma-functions in each product and the ambiguous signs are so chosen that the argument of each gamma-function contains one of the specified numbers of minus signs. Thenprovided that one of the numbers β, γ δ, ε is equal to ± n, where n is a positive integer; and the products and sums on the right range over the numbers α β, γ δ, ε.

Selection and Preparation of Specific Tissue Regions For Tem Using Large Epoxy-Embedded Blocks

1973 ◽  
Vol 31 ◽  
pp. 324-325 ◽  
Author(s):  
P. M. Lowrie ◽  
W. S. Tyler
Keyword(s):  
Electron Microscopy ◽  
Anatomical Structures ◽  
Ultrastructural Studies ◽  
Transmission Electron ◽  
Microscopic Evaluation ◽  
Embedded Blocks ◽  
Specific Tissue ◽  
Definition Of ◽  
Areas Of Interest ◽  
Selection Of

The importance of examining stained 1 to 2μ plastic sections by light microscopy has long been recognized, both for increased definition of many histologic features and for selection of specimen samples to be used in ultrastructural studies. Selection of specimens with specific orien ation relative to anatomical structures becomes of critical importance in ultrastructural investigations of organs such as the lung. The uantity of blocks necessary to locate special areas of interest by random sampling is large, however, and the method is lacking in precision. Several methods have been described for selection of specific areas for electron microscopy using light microscopic evaluation of paraffin, epoxy-infiltrated, or epoxy-embedded large blocks from which thick sections were cut. Selected areas from these thick sections were subsequently removed and re-embedded or attached to blank precasted blocks and resectioned for transmission electron microscopy (TEM).

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 Promotion of Resilience in Migrants: A Systematic Review of Study and Psychosocial Intervention

2021 ◽  
Author(s):  
Maria Ciaramella ◽  
Nadia Monacelli ◽  
Livia Concetta Eugenia Cocimano
Keyword(s):  
Systematic Review ◽  
Psychosocial Intervention ◽  
Life Experiences ◽  
Psychosocial Interventions ◽  
Migrant Population ◽  
Methodological Choices ◽  
Research Questions ◽  
Definition Of ◽  
Selection Of

AbstractThis systematic review aimed to contribute to a better and more focused understanding of the link between the concept of resilience and psychosocial interventions in the migrant population. The research questions concerned the type of population involved, definition of resilience, methodological choices and which intervention programmes were targeted at migrants. In the 90 articles included, an heterogeneity in defining resilience or not well specified definition resulted. Different migratory experiences were not adequately considered in the selection of participants. Few resilience interventions on migrants were resulted. A lack of procedure’s descriptions that keep in account specific migrants’ life-experiences and efficacy’s measures were highlighted.

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