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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 a 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 Cauchys residue theorem, and many known applications of the integral I(a, β , υ) are discussed by Leib-nitz’s rule for differentiation under the sign of integration. Further, one closed-form evaluation of the infinite series of the 1 F 0 (·) function is deduced. In addition, we establish some integral expressions in terms of the Euler numbers, which are not available in the Gradshteyn and Ryzhik book table. 2010 AMS Classification: 33C05, 33C20, 33B152010 AMS Classification: 33C05, 33C20, 33B15 |
