A Study of a Generalized Logarithmic–Hyperbolic Class of Integrals
Keywords:
Residue calculus, Mellin transform, Special functions, Definite integrals, Complex analysis, Fourier transform, Asymptotic expansions, logarithmic transformationsAbstract
We introduce a generalized class of definite integrals
I[f] = ∫(-∞ to ∞) f( ln(1 + x²) ) cosh(x) dx
where f is real- or complex-valued and satisfies mild growth conditions ensuring convergence. We obtain closed-form evaluations for several families of f, including polynomial/logarithmic choices, power-type families, and trigonometric functions of ln(1+x2). Full proofs of the intermediate lemmas and propositions are provided. Two complementary approaches are used: residue calculus for suitable meromorphic integrands and transform techniques based on Fourier and Mellin representations. We also derive asymptotic expansions for parameter families and include numerical verifications with illustrative plots. We conclude with open problems and natural generalizations.
Mathematics Subject Classification (2020): 26A42, 30E20, 44A10, 33B15, 41A60.
References
[1] Barreto DFM, Chesneau C. Study of a particular class of integrals. Int J Open Problems Comput Math. 2025, 18(4): 119–127.
[2] Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products, Academic Press, 7th/8th eds.
[3] Erdlyi A, et al. Tables of Integral Transforms, McGraw-Hill (Bateman Manuscript Project).
[4] Olver FWJ, et al. NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.
