Properties of Integrals Involving Ratios of the Modified Gamma Function
Abstract
In this article, we study the Modified Gamma function and more precisely we focus on properties of some integrals involving ratios of the Modified Gamma function. The properties studied involve estimates of square norms and Sobolev norms, using the definitions of L2 and Sobolev functional spaces. Additionally, estimates are derived where the integrand is the product of two or more functions involving particular ratios of the Modified Gamma function. Lastly, continuous entropy is computed for a particular function defined as ratio of the Modified Gamma function, and the corresponding continuous entropy is calculated for the derivative of the negative ratio of the previously mentioned function.
Downloads
References
Artin, E. (2015). The Gamma function. Dover Publications.
Andrews, G. E., Askey, R., & Ranjan, R. (1999). Special functions. Cambridge University Press. https://doi.org/10.1017/CBO9781107325937
Whittaker, E. T., & Watson, G. N. (2021). A course of modern analysis. Cambridge University Press. https://doi.org/10.1017/9781009004091
Lebedev, L. L. (1972). Special functions and their applications. Dover Publications.
Khan, D., Rehman, A., Iqbal, S., Ahmed, A., Jafar, S., & Baloch, M. (2021). Modified Fourier transform and its properties. Mathematica Montisnigri, 51, 60-73. https://doi.org/10.20948/mathmontis-2021-51-5
Ambarkane, N., Hatkar, S., & Bondar, K. (2023). Some results and applications of the exponential transform. Advances and Applications in Mathematical Sciences, 22(9), 2141-2154.
Ambarkhane, N. S., Dhirbasi, H. A., & Bondar, K. L. (2019). Exponential transform and its properties. International Journal of Mathematics Trends and Technology, 65(9), 84-93. https://doi.org/10.14445/22315373/IJMTT-V65I9P513
Wang, H., & Wu, Z. (2021). Certain m-convexity inequalities related to fractional integrals with exponential kernels. Open Access Library Journal, 8(4), e7388. https://doi.org/10.4236/oalib.1107388
Abrarov, S. M., Siddiqui, R., Jagpal, R. K., & Quine, B. M. (2021). A rational approximation of the Fourier transform by integration with exponential decay multiplier. Applied Mathematics, 12(11), 947-962. https://doi.org/10.4236/am.2021.1211063
Hilal, E. M. A. (2021). The differential transform. Journal of Applied Mathematics and Physics, 9(8), 1978-1992. https://doi.org/10.4236/jamp.2021.98129
Das, S. (2011). Functional fractional calculus (2nd ed.). Springer. https://doi.org/10.1007/978-3-642-20545-3
Kyriakis, A., & Kyriakis, M. (2024). Some inequalities involving the exponential transform. Advances in Pure Mathematics, 14(12), 859-872. https://doi.org/10.4236/apm.2024.1412047
Kyriakis, A. (2025). Some properties of the modified Gamma function. International Mathematical Forum, 20(2), 45-57. https://doi.org/10.12988/imf.2025.914486
This work is licensed under a Creative Commons Attribution 4.0 International License.
.jpg){kind=logo}
