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Analysis of Classical Special Beta & Gamma Functions in Engineering Mathematics and Physics

Authors

Pranesh Kulkarni
Type: Article
Publication Date: 2025-04-26
Citations: 0
DOI: https://doi.org/10.54105/ijam.a1195.05010425

Abstract

In many areas of applied mathematics, various types of Special functions have become essential tools for Scientists and engineers. Both Beta and Gamma functions are very important in calculus as complex integrals can be moderated into simpler form. In physics and engineering problems require a detailed knowledge of applied mathematics and an understanding of special functions such as gamma and beta functions. The topic of special functions is very important and it is constantly expanding with the existence of new problems in the applied Sciences in this article, we describe the basic theory of gamma and beta functions, their connections with each other and their applicability to engineering problems.to compute and depict scattering amplitude in Reggae trajectories. Our aim is to illustrate the extension of the classical beta function has many uses. It helps in providing new extensions of the beta distribution, providing new extensions of the Gauss hyper geometric functions and confluent hyper geometric function and generating relations, and extension of Riemann-Lowville derivatives. In this Article, we develop some elementary properties of the beta and gamma functions. We give more than one proof for some results. Often, one proof generalizes and others do not. We briefly discuss the finite field analogy of the gamma and beta functions. These are called Gauss and Jacobi sums and are important in number theory. We show how they can be used to prove Fermat's theorem that a prime of the form 4n + 1 is expressible as a sum of two squares. We also treat a simple multidimensional extension of a beta integral

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Locations

  • Indian Journal of Advanced Mathematics
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Special Functions

1999-01-13
George E. Andrews , Richard Askey , Ranjan Roy
Special functions, natural generalizations of the elementary functions, have been studied for centuries. The greatest mathematicians, among them Euler, Gauss, Legendre, Eisenstein, Riemann, and Ramanujan, have laid the foundations for … Special functions, natural generalizations of the elementary functions, have been studied for centuries. The greatest mathematicians, among them Euler, Gauss, Legendre, Eisenstein, Riemann, and Ramanujan, have laid the foundations for this beautiful and useful area of mathematics. This treatise presents an overview of special functions, focusing primarily on hypergeometric functions and the associated hypergeometric series, including Bessel functions and classical orthogonal polynomials, using the basic building block of the gamma function. In addition to relatively new work on gamma and beta functions, such as Selberg's multidimensional integrals, many important but relatively unknown nineteenth century results are included. Other topics include q-extensions of beta integrals and of hypergeometric series, Bailey chains, spherical harmonics, and applications to combinatorial problems. The authors provide organizing ideas, motivation, and historical background for the study and application of some important special functions. This clearly expressed and readable work can serve as a learning tool and lasting reference for students and researchers in special functions, mathematical physics, differential equations, mathematical computing, number theory, and combinatorics.

Calculation of Special Functions, the Gamma Function, the Exponential Integrals and Error-Like Functions (C. G. van der Laan and N. M. Temme)

1987-12-01
K. S. Kölbig
Previous article Next article Calculation of Special Functions, the Gamma Function, the Exponential Integrals and Error-Like Functions (C. G. van der Laan and N. M. Temme)K. S. KölbigK. S. Kölbighttps://doi.org/10.1137/1029138PDFBibTexSections … Previous article Next article Calculation of Special Functions, the Gamma Function, the Exponential Integrals and Error-Like Functions (C. G. van der Laan and N. M. Temme)K. S. KölbigK. S. Kölbighttps://doi.org/10.1137/1029138PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"Calculation of Special Functions, the Gamma Function, the Exponential Integrals and Error-Like Functions (C. G. van der Laan and N. M. Temme)." SIAM Review, 29(4), pp. 660–661[1] Milton Abramowitz and , Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964xiv+1046 29:4914 Google Scholar[2] Yudell L. Luke, The special functions and their approximations, Vol. I, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York, 1969xx+349 39:3039 0193.01701 Yudell L. Luke, The special functions and their approximations. Vol. II, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York, 1969xx+485 40:2909 0193.01701 Google Scholar[3] Yudell L. Luke, Mathematical functions and their approximations, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1975xvii+568 58:19039 0318.33001 CrossrefGoogle Scholar[4] Yudell L. Luke, Algorithms for the computation of mathematical functions, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1977xiii+284 58:13624 Google Scholar[5] O. I. Marichev, Handbook of integral transforms of higher transcendental functions, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1983, 336–, England 84f:00017 0494.33001 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Volume 29, Issue 4| 1987SIAM Review History Published online:17 February 2012 InformationCopyright © 1987 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1029138Article page range:pp. 660-661ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

General Properties of Generalized Gamma-Functions

2016-09-09
N. O. Virchenko
Background. The article is dedicated to studies of the main properties of new generalized gamma-functions, generalized incomplete gamma-functions, generalized digamma-functions for their best applications in applied sciences, for calculations of … Background. The article is dedicated to studies of the main properties of new generalized gamma-functions, generalized incomplete gamma-functions, generalized digamma-functions for their best applications in applied sciences, for calculations of integrals which are absent in scientific literature.Objective. Introduction and study of the basic properties of the new generalized gamma-functions, generalized incomplete gamma-functions, generalized digamma-functions and their applications.Methods. We apply the following methods: the methods of the theory of functions of the real variable, the theory of the special functions, the theory of the mathematical physics, the methods of applied analysis.Results. Some new forms of generalized gamma-functions, incomplete gamma-functions, digamma-functions are introduced. The main properties of these generalized special functions are explored. Examples of application of new generalized gamma-functions are given.Conclusions. With the help of the r-generalized confluent hypergeometric functions the new generalization of gamma-functions, incomplete gamma-functions, digamma-functions are introduced. The main properties of the new generalized special functions are explored, examples of application of these functions are given.
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The Generalized Incomplete Gamma Functions

2001-08-21
Engineering and physics demand a through knowledge of applied mathematics and a good understanding of special functions.These functions commonly arise in such areas of applications as heat conduction, communication systems, … Engineering and physics demand a through knowledge of applied mathematics and a good understanding of special functions.These functions commonly arise in such areas of applications as heat conduction, communication systems, electro-optics, approximation theory, probability theory, and electric circuit theory, among others.The subject of special functions is quite rich and expanding continuously with the emergence of new problems in the areas of applications in engineering and applied sciences.We investigate generalized gamma function, digamma function, the generalized incomplete gamma function, extended beta function.Also, some properties of these functions are taken into hand.
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Certain investigations in the field of special functions

2020-12-03
Maisoon Ahmed Kulib , Ahmed Ali Al-Gonah , Salem Saleh Barahmah
Motivated mainly by a variety of applications of Euler's Beta, hypergeometric, and confluent hypergeometric functions together with their extensions in a wide range of research fields such asengineering, chemical, and … Motivated mainly by a variety of applications of Euler's Beta, hypergeometric, and confluent hypergeometric functions together with their extensions in a wide range of research fields such asengineering, chemical, and physical problems. In this paper, we introduce modified forms of some extended special functions such as Gamma function, Beta function, hypergeometric function and confluent hypergeometric function by making use of the idea given in reference \cite{9}. Also, certain investigations including summation formulas, integral representations and Mellin transform of these modified functions are derived. Further, many known results are obtained asspecial cases of our main results.
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On a Class of Incomplete Gamma Functions with Applications

2001-08-21
M. Aslam Chaudhry , S. M. Zubair
The subject of special functions is rich and expanding continuously with the emergence of new problems encountered in engineering and applied science applications. The development of computational techniques and the … The subject of special functions is rich and expanding continuously with the emergence of new problems encountered in engineering and applied science applications. The development of computational techniques and the rapid growth in computing power have increased the importance of the special functions and their formulae for analytic representations

λ-Generalized Gamma Functions

2018-01-01
Asifa Tassaddiq
The objective of present research is to introduce and analyze a new special function that occurs as a part of the kernel of the integral representation of the λ-generalized Hurwitz-Lerch … The objective of present research is to introduce and analyze a new special function that occurs as a part of the kernel of the integral representation of the λ-generalized Hurwitz-Lerch zeta functions. One important aspect of the analysis of special functions is to study their properties. It is found that λ-generalized gamma function satisfies the log convex and derivative properties. Recurrence relation and reflection formula are achieved that are always important to study the behavior of new functions. As an application, a new series representation of the λ-generalized Hurwitz-Lerch zeta functions is established. These new results are validated by analyzing their important cases that agree with the known results. It is interesting to note that the coefficients in the series representation of the family of zeta functions are generalized from “1” to “gamma function” and then to “generalized gamma” and “λ-generalized gamma functions” in a simple and natural way.

Gamma and Beta Function Integrals

2020-01-01
Paul J. Nahin

Notes on Various Transforms Identified by Some Special Functions with Complex (or Real) Parameters and Some of Related Implications

2023-10-18
Hüseyi̇n Irmak , Tolga Han Açikgöz
The fundamental aim of this special research is first to introduce certain essential information in regards to some special functions, which are the Gamma function and the Beta function and … The fundamental aim of this special research is first to introduce certain essential information in regards to some special functions, which are the Gamma function and the Beta function and play a big role in both (applied) mathematics and most engineering sciences, and then to present both a number of their familiar properties and several relationships between them. Afterward, various possible-undeniable effects of those special functions in the transformation theory, their special implications, and suggestions for the relevant researchers will be also considered as special information.

A Study on Novel Extensions for the $p$-adic Gamma and $p$-adic Beta Functions

2019-05-21
Uğur Duran , Mehmet Açíkgöz
In this paper, we introduce the (ρ,q)-analogue of the p-adic factorial function. By utilizing some properties of (ρ,q)-numbers, we obtain several new and interesting identities and formulas. We then construct … In this paper, we introduce the (ρ,q)-analogue of the p-adic factorial function. By utilizing some properties of (ρ,q)-numbers, we obtain several new and interesting identities and formulas. We then construct the p-adic (ρ,q)-gamma function by means of the mentioned factorial function. We investigate several properties and relationships belonging to the foregoing gamma function, some of which are given for the case p = 2. We also derive more representations of the p-adic (ρ,q)-gamma function in general case. Moreover, we consider the p-adic (ρ,q)-Euler constant derived from the derivation of p-adic (ρ,q)-gamma function at x = 1. Furthermore, we provide a limit representation of aforementioned Euler constant based on (ρ,q)-numbers. Finally, we consider (ρ,q)-extension of the p-adic beta function via the p-adic (ρ,q)-gamma function and we then investigate various formulas and identities.
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The Gamma Function

2021-10-21
Charles H. C. Little , Kee L. Teo , Bruce van Brunt
The gamma function is a generalization of the factorial function. It is related to several other functions, including the trigonometric functions and the Riemann zeta function. This chapter is devoted … The gamma function is a generalization of the factorial function. It is related to several other functions, including the trigonometric functions and the Riemann zeta function. This chapter is devoted to the gamma function, functions that stem directly from the gamma function such as the digamma function, and applications of these functions.

q-Γάμμα συναρτήσεις

2015-07-02
Νίκη Παπαδήμα
Euler's Gamma function is one of the most basic special functions, not only in analysis but also in mathematical physics. The ongoing research in the area of mathematics and physics, … Euler's Gamma function is one of the most basic special functions, not only in analysis but also in mathematical physics. The ongoing research in the area of mathematics and physics, created the need to extend the gamma function. One of the extensions is the q-gamma function, which was a result of the introduction of q-calculus. In this paper, have been collected and recorded monotonicity properties and inequalities of functions involving q-gamma function. The first chapter lists the known properties of the gamma function. The second chapter presents the basic elements required of q calculus. The third chapter defines the q-gamma functions, q-Beta and q-ψ(x) and a reference is made to the properties that apply to them. In the fourth chapter are presented properties of the monotonicity of functions that contain q-gamma functions and inequalities that these functions satisfy. The results, recorded, are assembled from papers that have been published, related to the q-gamma functions and many of these generalize results related to gamma functions

NOTE ON CONVERGENCE OF EULER'S GAMMA FUNCTION

2013-03-25
Junesang Choi
The Gamma function <TEX>${\Gamma}$</TEX> which was first introduced b Euler in 1730 has played a very important role in many branches of mathematics, especially, in the theory of special functions, … The Gamma function <TEX>${\Gamma}$</TEX> which was first introduced b Euler in 1730 has played a very important role in many branches of mathematics, especially, in the theory of special functions, and has been introduced in most of calculus textbooks. In this note, our major aim is to explain the convergence of the Euler's Gamma function expressed as an improper integral by using some elementary properties and a fundamental axiom holding on the set of real numbers <TEX>$\mathbb{R}$</TEX>, in a detailed and instructive manner. A brief history and origin of the Gamma function is also considered.
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SPECIAL FUNCTIONS: GAMMA & ZETA

2012-01-01
Josh Engwer
n=1 ( 1 + z n )−1 e where the constant γ ≈ 0.57722 is chosen s.t. Γ(1) = 1. Note that Γ(z) has simple poles at z ∈ Z− … n=1 ( 1 + z n )−1 e where the constant γ ≈ 0.57722 is chosen s.t. Γ(1) = 1. Note that Γ(z) has simple poles at z ∈ Z− • Gauss’s Formula for the Gamma Function : Γ(z) = lim n→∞ n!n z(z + 1) · · · (z + n) ∀z ∈ C Z− • Functional Equation for the Gamma Function : Γ(z + 1) = zΓ(z) ∀z ∈ C Z− • Relationship between the Gamma Function and the Factorial : Γ(n+ 1) = n! ∀n ∈ Z+ • Relationship between the Gamma Function and the Cosecant : Γ(z)Γ(1− z) = π csc(πz) ∀z ∈ C Z • Integral Form of the Gamma Function : Γ(z) = ∫ ∞

Investigation of the k-Analogue of Gauss Hypergeometric Functions Constructed by the Hadamard Product

2021-04-18
Mohamed Abdalla , Muajebah Hidan
Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the … Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the Pochhammer symbol, and hypergeometric functions, this work is devoted to the study of the k-analogue of Gauss hypergeometric functions by the Hadamard product. We give a definition of the Hadamard product of k-Gauss hypergeometric functions (HPkGHF) associated with the fourth numerator and two denominator parameters. In addition, convergence properties are derived from this function. We also discuss interesting properties such as derivative formulae, integral representations, and integral transforms including beta transform and Laplace transform. Furthermore, we investigate some contiguous function relations and differential equations connecting the HPkGHF. The current results are more general than previous ones. Moreover, the proposed results are useful in the theory of k-special functions where the hypergeometric function naturally occurs.
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A new definition of generalization of the factorial function with new results about the gamma of negative integers

2023-09-18
Mohammed Hussein
Abstract The gamma function is a mathematical function that generalizes the concept of factorial to real and complex numbers. While the gamma function is a powerful tool in mathematics, it … Abstract The gamma function is a mathematical function that generalizes the concept of factorial to real and complex numbers. While the gamma function is a powerful tool in mathematics, it does have certain limitations and potential issues for example, · Non-integer values: The gamma function is not defined for negative integers, this limitation can be problematic in certain contexts where negative integer values are involved. · Pole at zero: The gamma function has a pole at zero, which means it is undefined at this point. This can pose challenges when working with functions or equations that involve the gamma function near or at zero. · Computational complexity: Computing the gamma function numerically can be computationally expensive and time-consuming, especially for large or complex arguments. In this study, we have addressed the aforementioned issues by proposing a new definition for generalizing the factorial function, which serves as an alternative definition of the gamma function. This new definition is formulated based on the utilization of the differential operator. The resulting definition stands out for its simplicity and effectiveness in computing real numbers, including non-positive integers. Moreover, our research has yielded fresh insights into the gamma function's behavior with respect to non-positive integers, potentially leading to a transformative approach in employing fractional differential and integral equations to describe a wide range of cosmic phenomena.
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Expression of Some Special Functions through &amp;lt;i&amp;gt;q&amp;lt;/i&amp;gt;-Exponentials of the Nonadditive Statistical Mechanics

2020-01-01
L.S. Lima
Generalized q-exponentials functions are employed to make a generalization of complete and incomplete gamma functions. We obtain a generalization of this class of special functions which are very important in … Generalized q-exponentials functions are employed to make a generalization of complete and incomplete gamma functions. We obtain a generalization of this class of special functions which are very important in the fields of probability, statistics, statistical physics as well as combinatorics and we derive some of its properties. One gets that the generalized gamma function obtained whether approaches of the standard gamma function for a specific q values such as q=q0≈0.9 value suffering a large variation with the variation of q.
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A new definition of generalization of the factorial function with new results about the gamma of negative integers

2023-09-14
Mohammed Hussein
Abstract The gamma function is a mathematical function that generalizes the concept of factorial to real and complex numbers. While the gamma function is a powerful tool in mathematics, it … Abstract The gamma function is a mathematical function that generalizes the concept of factorial to real and complex numbers. While the gamma function is a powerful tool in mathematics, it does have certain limitations and potential issues for example, • Non-integer values: The gamma function is not defined for negative integers, this limitation can be problematic in certain contexts where negative integer values are involved. • Pole at zero: The gamma function has a pole at zero, which means it is undefined at this point. This can pose challenges when working with functions or equations that involve the gamma function near or at zero. • Computational complexity: Computing the gamma function numerically can be computationally expensive and time-consuming, especially for large or complex arguments. In this study, we have addressed the aforementioned issues by proposing a new definition for generalizing the factorial function, which serves as an alternative definition of the gamma function. This new definition is formulated based on the utilization of the differential operator. The resulting definition stands out for its simplicity and effectiveness in computing real numbers, including non-positive integers. Moreover, our research has yielded fresh insights into the gamma function's behavior with respect to non-positive integers, potentially leading to a transformative approach in employing fractional differential and integral equations to describe a wide range of cosmic phenomena.
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ANALYSIS OF NOTES ON THE THEORY OF GAMMA FUNCTION FROM NOTEBOOKS OF L. EULER

2020-02-17
И.В. Игнатушина , Olexandr Polishchuk
Великому ученому Леонарду Эйлеру (17071783), одному из основоположников современной математики, принадлежит решающая роль в формировании некоторых разделов теории специальных функций. В частности, им были введены понятия функций гамма (Г) и … Великому ученому Леонарду Эйлеру (17071783), одному из основоположников современной математики, принадлежит решающая роль в формировании некоторых разделов теории специальных функций. В частности, им были введены понятия функций гамма (Г) и бета (В) и получены важные результаты относительно свойств этих функций, которые находят широкое применение в различных отраслях современной науки. В предлагаемой статье представлен анализ страниц из записных книжек Эйлера, касающихся теории гамма-функции. The great scientist Leonard Euler (17071783), one of the founders of modern mathematics, has a decisive role in the formation of some sections of the theory of special functions. In particular, he introduced the concepts of gamma (G) and beta (B) functions and obtained important results regarding the properties of these functions, which are widely used in various branches of modern science. This article presents an analysis of pages from Eulers notebooks regarding the theory of gamma function.

Investigation of Integral Transformation Associated with Extended Generalized Srivastava’s Hypergeometric Multi Variable Special Function

2024-03-22
Mausmi Kulmitra , Surendra Kumar Tiwari
In recent year study on multivariate special functions and Integral transformation have been booming. In this work, we have focused on Srivastava hypergeometric function , , and with triple variable. … In recent year study on multivariate special functions and Integral transformation have been booming. In this work, we have focused on Srivastava hypergeometric function , , and with triple variable. We have discussed the literature study and motivation from the recent works on the extension of Srivastava’s multivariable hypergeometric function , , and . In this paper, the extension of , , and is studied based on the generalized beta function and the generalized Pochhammer’s symbol . Furthermore, the Mellin integral transformation and Inverse Mellin integral transformation have been studied for the based extension of the functions , , and . A few of the most recent uses of these transformations in various scientific and engineering fields are also highlighted in this paper. In general, this work seeks to offer a thorough overview of recent breakthroughs in the importance and applications of several integral transforms of Multivariable functions.
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