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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 13

The Krätzel Function as the H-Function and the Evaluation of Integrals

A.A. Kilbas1, M. Saigo2, R.K. Saxena3 and J.J. Trujillo4

1Department of Mathematics and Mechanics, Belarusian State University, Belarus
2Department of Applied Mathematics, Fukuoka University, Japan
3Department of Mathematics and Statistics, Jai Narain Vyas University, India
4Departamento de Análisis Matemático, Universidad de la Laguna, Spain

Full Bibliographic Reference for this paper
A.A. Kilbas, M. Saigo, R.K. Saxena, J.J. Trujillo, "The Krätzel Function as the H-Function and the Evaluation of Integrals", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 13, 2006. doi:10.4203/ccp.84.13
Keywords: Krätzel function, asymptotic estimates, evaluation of integrals, H-function, Meijer G-function, Bessel and hypergeometric functions, Mellin transform.

Summary
The paper is devoted to the study of the function defined for , and , being such that when , by
    (26)

This function, coinciding for and with the McDonald function apart from the power function:
    (27)

for was introduced by Krätzel. Properties of the function (26) with were investigated by some authors. In particular, the explicit solutions of some ordinary and partial differential equations of fractional order were obtained in terms of . Such investigations now are of a great interest in connection with applications,

We establish the formula for the Mellin transform of :

    (28)

provided that , and are such that when , while when . Using (28) we extend from positive to complex z by means of its representation in terms of the H-function in the form
    (29)

when while for
    (30)

and give conditions for these representations.

Note that for integers such that and for and the function is defined via a Mellin-Barnes integral in the form

(31)


    (32)

An empty product in (32), if it occurs, is taken to be one. in (31) is an infinite contour which separates all poles of the Gamma functions to the left and all poles of the Gamma functions to the right of .

Using (29) and (30), we find the asymptotic estimates for at zero and infinity. Our results in the case give more precisely the known asymptotic relations for as and as

We apply the obtained results to evaluate the integral

    (33)

with , and involving the function and the H-function in the integrand. We prove that the integral (31) is the H-function of the form and for and respectively. The special cases of such formulas are presented when coincides with the Meijer G-function. Applications are given to evaluate integrals involving a product of and the Bessel function of the first kind , the McDonald function , the Whittaker confluent hypergeometric function and the Gauss hypergeometric function . Special cases involving products of and elementary functions are also presented.

It should be noted that there are many known integrals of special functions that can be evaluated in terms of elementary and special functions. The integrals evaluated in this paper are new.

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