Abstract
In this paper, we investigate and evaluate the analytical expressions for some definite integrals of Srinivasa Ramanujan in terms of Meijer’s G-function by using the Laplace transforms of
Funding source: University Grants Commission
Award Identifier / Grant number: F.4-2/2006 (BSR)/MA/20-21/0061
Funding statement: This study was funded by the University grants commission of India for the award of a Dr. D. S. Kothari Post Doctoral Fellowship (DSKPDF) (grant number F.4-2/2006 (BSR)/MA/20-21/0061).
References
[1] B. C. Berndt, Ramanujan’s Notebooks. Part IV, Springer, New York, 1994. 10.1007/978-1-4612-0879-2Search in Google Scholar
[2] K. N. Boyadzhiev and V. H. Moll, The integrals in Gradshteyn and Ryzhik. Part 21: Hyperbolic functions, Sci. Ser. A Math. Sci. (N. S.) 22 (2012), 109–127. Search in Google Scholar
[3] M. W. Coffey, Integrals in Gradshteyn and Rhyzhik: Hyperbolic and trigonometric integrals, preprint (2018), https://arxiv.org/abs/1803.00632. Search in Google Scholar
[4] S. A. Dar and R. B. Paris, On integrals involving quotients of hyperbolic functions, J. Ramanujan Math. Soc. 36 (2021), no. 1, 23–32. Search in Google Scholar
[5] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. Vol. 1, McGraw-Hill, New York, 1953. Search in Google Scholar
[6] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms. Vol. 1, McGraw-Hill, New York, 1954. Search in Google Scholar
[7] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 8th ed., Elsevier/Academic, Amsterdam, 2015. Search in Google Scholar
[8] G. H. Hardy, On a class of definite integrals containing hyperbolic functions, Mess. Math. 29 (1900), 25–42. Search in Google Scholar
[9] A. M. Mathai and R. K. Saxena, Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Lecture Notes in Math. 348, Springer, Berlin, 1973. 10.1007/BFb0060468Search in Google Scholar
[10] A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines, John Wiley & Sons, New York, 1978. Search in Google Scholar
[11] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University, Cambridge, 2010. Search in Google Scholar
[12] M. I. Qureshi and S. A. Dar, Evaluation of some definite integrals of Ramanujan, using hypergeometric approach, Palest. J. Math. 7 (2018), no. 2, 620–623. Search in Google Scholar
[13] M. I. Qureshi and S. A. Dar, Generalizations of Ramanujan’s integral associated with infinite Fourier cosine transforms in terms of hypergeometric functions and its applications, Kyungpook Math. J. 60 (2020), no. 4, 781–795. Search in Google Scholar
[14] M. I. Qureshi and S. A. Dar, Computation of three theorems of Srinivasa Ramanujan associated with definite integrals connected with Gauss sums, Palest. J. Math. 10 (2021), no. 1, 184–198. Search in Google Scholar
[15]
M. I. Qureshi and S. A. Dar,
Generalizations and applications of Srinivasa Ramanujan’s integral
[16] S. Ramanujan, Some definite integrals connected with Gauss’s sums., Mess. Math. 44 (1915), 75–86. Search in Google Scholar
[17] S. Ramanujan, Collected Papers of Srinivasa Ramanujan, AMS Chelsea, Providence, 2000. Search in Google Scholar
[18] H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Two Variables. With Applications, South Asian Publishers, New Delhi, 1982. Search in Google Scholar
[19] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Ellis Horwood Ser. Math. Appl., Ellis Horwood, Chichester, 1984. Search in Google Scholar
[20] H. M. Srivastava, M. I. Qureshi and S. A. Dar, Some novel Laplace-transform based integrals via hypergeometric techniques, Appl. Math. Inf. Sci. 14 (2020), no. 5, 743–754. 10.18576/amis/140501Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Existence and uniqueness of solutions to higher order fractional partial differential equations with purely integral conditions
- Stabilization by an estimated state controller of nonlinear time-varying systems
- Global existence and blow-up of solutions for coupled bi-harmonic nonlinear wave equations
- Consequences of Srinivasa Ramanujan integrals involving Meijer’s G-function
- Parameterized Simpson-like inequalities for differential s-convex functions
Articles in the same Issue
- Frontmatter
- Existence and uniqueness of solutions to higher order fractional partial differential equations with purely integral conditions
- Stabilization by an estimated state controller of nonlinear time-varying systems
- Global existence and blow-up of solutions for coupled bi-harmonic nonlinear wave equations
- Consequences of Srinivasa Ramanujan integrals involving Meijer’s G-function
- Parameterized Simpson-like inequalities for differential s-convex functions