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Consequences of an infinite Fourier cosine transform-based Ramanujan integral

  • Showkat Ahmad Dar ORCID logo EMAIL logo , Mohammad Kamarujjama , W. M. Shah and Daud ORCID logo
Published/Copyright: January 11, 2024

Abstract

In this paper, we express a generalization of the Ramanujan integral I ( α ) with the analytical solutions, using the Laplace transform technique and some algebraic relation or the Pochhammer symbol. Moreover, we evaluate some consequences of a generalized definite integral ϕ * ( υ , β , a ) . The well-known special cases appeared, whose solutions are possible by Cauchy’s residue theorem, and many known applications of the integral I ( a , β , υ ) are discussed by the Leibniz rule for differentiation under the sign of integration. Further, one closed-form evaluation of the infinite series of the F 0 1 ( ) function is deduced. In addition, we establish some integral expressions in terms of the Euler numbers, which are not available in the tables of the book of Gradshteyn and Ryzhik.

MSC 2020: 33C05; 33C20; 33B15

Award Identifier / Grant number: F.4-2/2006

Funding statement: This study was funded by 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).

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Received: 2023-07-11
Accepted: 2023-11-29
Published Online: 2024-01-11
Published in Print: 2024-11-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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