Products of Incomplete gamma functions Integral representations

Year 2016, Volume: 4 Issue: 2, 47 - 51, 30.10.2016

Rami Alahmad

https://doi.org/10.36753/mathenot.421452

Cited By: 1

Abstract

In this paper we find integral representations, involving incomplete gamma and incomplete beta functions,
of products of incomplete gamma functions. Also, in this paper we find interesting relations between
incomplete gamma functions and Laplace transform. Since the error function is an incomplete gamma
function, we find interesting relations between error functions and Laplace transform. Using the results
above we find several interesting integrals.

Keywords

: Incomplete gamma functions, Incomplete beta functions, error functions, Laplace transform

References

• [1] Agarwal, P. and Nieto, J. J., Some fractional integral formulas for the Mittag-Leffer type function with four parameters. Open Mathematics 13 (2015), no. 1, 537–546.
• [2] Agarwal, P., Chand, M. and Jain, S., Certain integrals involving generalized Mittag-Leffer functions. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 85 (2015), no. 3, 359–371.
• [3] Agarwal, P. and Jain, S., A new class of integral relations involving a general class of polynomials and I-functions. Walailak J Sci & Tech. 12 (2015), no. 11, 1009–1018.
• [4] AlAhmad, R., Products of incomplete gamma functions. Analysis ISSN (Online) 2196-6753, ISSN (Print) 0174- 4747, DOI: 10.1515/anly-2015-0012 (2015).
• [5] Apelblat, A., Table of definite and infinite integrals. Physical Sciences Data, Elsevier Scientific Publishing Co., Amsterdam, 1983.
• [6] Çetinkaya, A., The incomplete second Appell hypergeometric functions. Applied Mathematics and Computation 219 (2013), 8332–8337.
• [7] Choi, J. and Agarwal, P., Certain integral transforms for the incomplete functions.Appl. Math. 9 (2015), no. 4, 2161–2167.
• [8] Gradshteyn ,I. S. and Ryzhik ,I. M., Tables of integrals, series and products. Academic Press, New York, 1980.
• [9] Metzler, R., Klafter, J. and Jortner, J., Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. 96 (1999), no. 20, 11085–11089.
• [10] Miller, A. R. and Moskowitz, I. S., On certain generalized incomplete gamma functions. J. Comp. Appl. Math. 91 (1963), no. 2, 179–190.
• [11] Shavitt, S., The gaussian function in calculations of statistical mechanics and quantum mechanics, chapter in methods in computational physics. Academic Press, New York, 1963.
• [12] Shavitt, I. and Karplus, M., Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J. Chem. Phys. 43 (1965), no. 2, 398–414.
• [13] Sornette, D., Multiplicative processes and power laws. Phys. Rev. E. 57 (1998), no. 4, 4811–4813.
• [14] Srivastava, H. M. and Agarwal, P., Certain fractional integral operators and the generalized incomplete hypergeometric functions. Applications and Applied Mathematics, 2013.
• [15] Temme, N., Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function. Math. Comp. 29 (1975), no. 132, 1109–1114