A Generalization of Ramanujan-Type Integrals

Irshad Ayoob

International Journal of Mathematics and Mathematical Sciences, 2026, vol. 2026, 1-10

Abstract: Recently, Xu, Yin, and Zhou obtained explicit evaluations of the Ramanujan-type integrals ∫0∞xs−1sinbx/coshax−cosbxdx and ∫0∞xs−1sinbx/coshax+cosbxdx, expressing them in closed form in terms of the gamma function, the Riemann zeta function, and the alternating zeta function. In the present paper, we generalize these formulas from the special Mellin kernel xs−1 to a general real-valued measurable function f and evaluate the two integral transforms ∫0∞fx sinbx/coshax−cosbxdx and ∫0∞fxsinbx/coshax+cosbxdx, where a>0, b∈R, and f satisfies suitable Laplace-transform and summability assumptions. If Fz=∫0∞fxe−zx dx denotes the Laplace transform of f, then we prove the exact representations ∫0∞fxsinbx/coshax−cosbxdx=2∑n=1∞IFna−ib, and ∫0∞fxsinbx/coshax+cosbxdx=2∑n=1∞−1n+1IFna−ib. Thus, our results may be viewed as functional extensions of Xu’s formulas. As applications, we recover the known Mellin-type evaluations by taking fx=xs−1, and we derive further explicit corollaries for fx=xs−1e−cx and fx=xe−cx, leading respectively to identities involving special functions.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:5551612

DOI: 10.1155/ijmm/5551612

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