scholarly journals Mellin–Barnes integrals and the method of brackets

2022 ◽  
Vol 82 (1) ◽  
Author(s):  
Ivan Gonzalez ◽  
Igor Kondrashuk ◽  
Victor H. Moll ◽  
Luis M. Recabarren
Keyword(s):  
Gamma Function ◽  
Main Part ◽  
Integral Representations ◽  
Fundamental Idea ◽  
Definite Integrals

AbstractThe method of brackets is a method for the evaluation of definite integrals based on a small number of rules. This is employed here for the evaluation of Mellin–Barnes integral. The fundamental idea is to transform these integral representations into a bracket series to obtain their values. The expansion of the gamma function in such a series constitute the main part of this new application. The power and flexibility of this procedure is illustrated with a variety of examples.

scholarly journals Integral representations of quantities associated with Gamma function

2021 ◽  
Vol 13 (4) ◽  
pp. 50-62
Author(s):  
Andrew Borisovich Kostin ◽  
Vladimir Borisovich Sherstyukov
Keyword(s):  
Gamma Function ◽  
Integral Representations

scholarly journals Regularized Integral Representations of the Reciprocal Gamma Function

2019 ◽  
Vol 3 (1) ◽  
pp. 1 ◽  
Author(s):  
Dimiter Prodanov
Keyword(s):  
Integral Representation ◽  
Computer Algebra ◽  
Gamma Function ◽  
Real Line ◽  
Computer Algebra System ◽  
Integral Representations ◽  
Complex Representation ◽  
Hypersingular Integral ◽  
The Real ◽  
Two Parameter

This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine’s complex representation is demonstrated. For both real and complex integrals, the regularized representation can be expressed in terms of the two-parameter Mittag-Leffler function. Reference numerical implementations in the Computer Algebra System Maxima are provided.

VARIOUS INTEGRAL REPRESENTATIONS OF GAMMA FUNCTION

2015 ◽  
Vol 92 (1) ◽  
pp. 31-49
Author(s):  
Mian Arif Shams Adnan ◽  
Humayun Kiser ◽  
Zahida Sultana Irin ◽  
Asif Shams Adnan ◽  
M. Shamsuddin
Keyword(s):  
Gamma Function ◽  
Integral Representations

scholarly journals Extended k-Gamma and k-Beta Functions of Matrix Arguments

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1715
Author(s):  
Ghazi S. Khammash ◽  
Praveen Agarwal ◽  
Junesang Choi
Keyword(s):  
Gamma Function ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Integral Formula ◽  
Beta Function ◽  
Integral Representations ◽  
Matrix Argument ◽  
Functional Relations

Various k-special functions such as k-gamma function, k-beta function and k-hypergeometric functions have been introduced and investigated. Recently, the k-gamma function of a matrix argument and k-beta function of matrix arguments have been presented and studied. In this paper, we aim to introduce an extended k-gamma function of a matrix argument and an extended k-beta function of matrix arguments and investigate some of their properties such as functional relations, inequality, integral formula, and integral representations. Also an application of the extended k-beta function of matrix arguments to statistics is considered.

scholarly journals On integral representations of the gamma function

2021 ◽  
Author(s):  
Андрей Костин ◽  
Владимир Шерстюков
Keyword(s):  
Gamma Function ◽  
Integral Representations

Generalizations and applications of Srinivasa Ramanujan’s integral associated with infinite Fourier sine transforms in terms of Meijer’s G-function

Analysis ◽  
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammad Idris Qureshi ◽  
Showkat Ahmad Dar
Keyword(s):  
Laplace Transform ◽  
Analytical Solutions ◽  
Integral Representations ◽  
Sine Function ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Convergence Conditions ◽  
Definite Integrals ◽  
Summation Formulas ◽  
Algebraic Properties

Abstract In this paper, we obtain analytical solutions of an unsolved integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} of Srinivasa Ramanujan [S. Ramanujan, Some definite integrals connected with Gauss’s sums, Mess. Math. 44 1915, 75–86] with suitable convergence conditions in terms of Meijer’s G-function of one variable, by using Mellin–Barnes type contour integral representations of the sine function, Laplace transform method and some algebraic properties of Pochhammer’s symbol. Also, we have given some generalizations of Ramanujan’s integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} in the form of integrals ℧ S * ⁢ ( υ , b , c , λ , y ) {\mho_{S}^{*}(\upsilon,b,c,\lambda,y)} , Ξ S ⁢ ( υ , b , c , λ , y ) {\Xi_{S}(\upsilon,b,c,\lambda,y)} , ∇ S ⁡ ( υ , b , c , λ , y ) {\nabla_{S}(\upsilon,b,c,\lambda,y)} and ℧ S ⁢ ( υ , b , λ , y ) {\mho_{S}(\upsilon,b,\lambda,y)} with suitable convergence conditions and solved them in terms of Meijer’s G-functions. Moreover, as applications of Ramanujan’s integral 𝐑 S ⁢ ( m , n ) {\mathbf{R}_{S}(m,n)} , the three new infinite summation formulas associated with Meijer’s G-function are obtained.

scholarly journals Extended Mellin integral representations for the absolute value of the gamma function

Analysis ◽  
2018 ◽  
Vol 38 (1) ◽  
pp. 11-20
Author(s):  
Nicolas Privault
Keyword(s):  
Fourier Transform ◽  
Gamma Function ◽  
Bessel Functions ◽  
Planck Equation ◽  
Integral Representations ◽  
Modified Bessel Functions ◽  
Absolute Value ◽  
Fokker Planck Equation ◽  
The Absolute ◽  
Fokker Planck

AbstractWe derive Mellin integral representations in terms of Macdonald functions for the squared absolute value{s\mapsto|\Gamma(a+is)|^{2}}of the gamma function and its Fourier transform when{a<0}is non-integer, generalizing known results in the case{a>0}. This representation is based on a renormalization argument using modified Bessel functions of the second kind, and it applies to the representation of the solutions of a Fokker–Planck equation.

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