|
|||||||
|
Evaluation of General Class of
Beta Integrals Involving Certain Particular Functions |
|||||||
| Paper Id :
18500 Submission Date :
08/02/2024 Acceptance Date :
15/02/2024 Publication Date :
20/02/2024
This is an open-access research paper/article distributed under the terms of the Creative Commons Attribution 4.0 International, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. DOI:10.5281/zenodo.10673412 For verification of this paper, please visit on
http://www.socialresearchfoundation.com/innovation.php#8
|
|||||||
| |||||||
| Abstract |
In this paper general class of Beta integrals is considered
for particular function |
||||||
|---|---|---|---|---|---|---|---|
| Keywords | Beta Function, Gamma Function, Exponential Functions, Gauss’s Hypergeometric Function, Extended Beta Function, Generalized Hypergeometric Function, Hurwitz-Lerch zeta Function.. | ||||||
| Introduction | The well-known Gamma and Beta function are defined and represented by following integrals.[10]
Choudhary et. al. [1][2] studied and extended these function to the entire complex plane by inserting the
|
||||||
| Aim of study |
We have studied general class of Beta integrals for
particular functions in. This class is used to evaluate certain integrals for
some special functions by assigning the different- different forms of the
functions. Various known and new simpler integral formulae are obtained from
our main results. As the application of these simpler results many interesting
consequences are derived. The methodology and technique that we have used in
this paper to obtain the results can be further extended for more general new
functions. The young researchers can use our finding to develop new study. |
||||||
| Review of Literature | The Beta function is a unique function
where it is classified as the first
kind of Euler’s integral. The Beta function is defined in the
domains of real numbers. The notation to represent the beta function is “β”. The beta function
is meant by B(p, q), where the parameters p and q should be real numbersThe
Beta function was introducedearly 19th century.The
function was later extensively studied by other mathematicians, including
Augustin-Louis Cauchy and CarlGustav Jacobi, who contributed to understanding
its propertiesand relationship with
other mathematical functions. The Gamma function was first introduced by
Swiss mathematicianLeonhard Euler in the 18th centuryalthough
it was later formalized and studied extensively by other mathematicians Both functions have since been further
developed and studied by numerous mathematicians and have found applications in
variousareas ofMathematics, Physics and engineering.Due
to usefulness of these functions Chaudhary and Zubair studiedextended Gamma and
Beta functions The Pochhammer symbol, also known as the
rising factorial, was named after the German mathematician Leo August
Pochhammer. Pochhammer lived from 1841 to 1920 and made significant
contributions to various areas of mathematics, including number theory,
algebra, and analysis. The Gauss hypergeometric function 2F1
is a special function that plays a fundamental role in many areas of
mathematics, physics, and engineering. It is named after the German
mathematician Carl Friedrich Gauss.The Gauss hypergeometric function has
numerous properties and relationships with other mathematical functions, and
its study continues to be an active area of research in mathematics and its
applications. F1, F2,
F3, and F4 are the
functions that Appell developed in 1880, now popularly known as Appell
functions. These are two variables extension of 2F1 function. |
||||||
| Result and Discussion | It is useful here to define the following relations for Pocchammer symbol[10], pp.21-23; eqs. (4), (15),
Main Result In this section we will study the following general class of integrals in view of the series integral representation of extended Beta function in (1.7)
|
||||||
| Conclusion |
|
||||||
| References | 1. Chaudhary, M.A.,
Quadir, A.Rafique, M. And Zubair, S.M., Extension of Euler’s beta function,
J.comput. Appl. Math78(1997) 19-32 2. Chaudhary, M.A.,
Zubair, S., Generalized incomplete gamma functions with applications, J.comput.
Appl. Math55 (1994) 99-124 3. Ismail, M.E.H. and
Pitman, J., Algebraic evaluation of some Euler integrals, duplication formulae
for Appell’s hypergeometric function F1 and Brownian variations, C and J. Math
52(5) (2000), 961-981. 4. Jaimini B.B. and
Sharma, M., Some results on extended Beta function. J.Raj. Acad. Phy. Sci. Vol.7,
No. 3,(2008), 351-358. 5. Khan S., Agarwal,
B., Pathan, M.A. and Mohammad F., Evaluation of Certain Euler type integrals,
J. Appl. Math and Comp. 189 (2007) 1993-2003. 6. Lin, S.D., Tu,
S.T., Haries, T.M. and Srivastava H.M., Some finite and infinite sums
associated with Digamma and related functions. J. Fract. Calc., 22(2002),
103-114. 7. Nishimoto, K., Yen,
C.E. and Lin, M.L., Some integral forms of a generalized Zeta functions, J.
Fract. Calc. 22 (2002), 91-97. 8. Pathan, M.A.,
Jaimini, B.B. and Gautam S., On a general class of Euler type integrals Proc.
Int. Conf. SSFA Vol.8 (2007), 221-232. 9. Pathan, M.A., Jaimini, B.B., and Sharma, M., On evaluation of
certain general classes of Euler Type
integrals Proc. Int. Comf. SSFA, Vol.8 (2007) 221-232. 10. Rainville E.D.,
Special functions, Chelsea Publishing Co., Bronx, New York (1960). 11. Srivastava, H.M.
and Karlsson, P.W., Multiple Gaussian hypergeometric series, Ellis Horwood
Ltd., Chichester, (1985). |
||||||
