{"paper":{"title":"A Generalized Closed Form of Ramanujan-Type Fourier Cosine Transform via Meijer's G-Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The Ramanujan integral R_C(m,n) equals an infinite series of Meijer G-functions under convergence conditions on m and n.","cross_cats":[],"primary_cat":"math.NT","authors_text":"R. P. Paris, S. A. Dar","submitted_at":"2026-05-11T08:10:36Z","abstract_excerpt":"In this paper, we obtain analytical evaluations of the Ramanujan integral \\[\\textbf{R}_{C}(m,n)= \\int_{0}^{\\infty}\\frac{x^m\\,\\cos(\\pi nx)}{\\exp{(2\\pi\\sqrt{x})-1}}dx\\] subject to suitable convergence conditions in terms of an infinite series of Meijer $G$-functions of one variable, by using Mellin-Barnes-type contour integral representations of the cosine function. %and Laplace transform method. We also consider some generalizations of the integral $\\textbf{R}_{C}(m,n)$ given as the integrals $I_{C}^{*}(\\upsilon,b,c,\\lambda,y)$ ,$\\Xi_{C}(\\upsilon,b,c,\\lambda,y)$, $\\nabla_{C}(\\upsilon,b,c,\\lambd"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We obtain analytical evaluations of the Ramanujan integral R_C(m,n) subject to suitable convergence conditions in terms of an infinite series of Meijer G-functions of one variable, by using Mellin-Barnes-type contour integral representations of the cosine function.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The interchange of integration order and the absolute convergence of the resulting contour integrals under the stated conditions on m and n.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Ramanujan integral R_C(m,n) equals an infinite series of Meijer G-functions, with generalizations and closed forms for nine related series.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Ramanujan integral R_C(m,n) equals an infinite series of Meijer G-functions under convergence conditions on m and n.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b5ad7b59a663c7989c209bec4615a239b55769a02d0b912b8a160aa61cc6ea99"},"source":{"id":"2605.13882","kind":"arxiv","version":1},"verdict":{"id":"4e821ee8-7eb1-4d2b-b0ae-ba52d5d3fd13","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T05:57:50.950038Z","strongest_claim":"We obtain analytical evaluations of the Ramanujan integral R_C(m,n) subject to suitable convergence conditions in terms of an infinite series of Meijer G-functions of one variable, by using Mellin-Barnes-type contour integral representations of the cosine function.","one_line_summary":"The Ramanujan integral R_C(m,n) equals an infinite series of Meijer G-functions, with generalizations and closed forms for nine related series.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The interchange of integration order and the absolute convergence of the resulting contour integrals under the stated conditions on m and n.","pith_extraction_headline":"The Ramanujan integral R_C(m,n) equals an infinite series of Meijer G-functions under convergence conditions on m and n."},"references":{"count":20,"sample":[{"doi":"","year":2021,"title":"C., and Straub, A","work_id":"2a767044-e0a1-4252-8e43-4a014994e3a6","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"Berndt, B. C. ; Integrals associated with Ramanujan and e lliptic functions. The Ramanujan Journal, 1, 2016","work_id":"3dcc1c79-3f3a-4218-93a2-0666615d422a","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1921,"title":"Carslaw, H. S. ; Introduction to the theory of F ourier’s series and integral s. Macmillan and co., limited st. Martin’s street, London, 1921","work_id":"e26043db-2b25-446c-b602-0e2f32624e29","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1965,"title":"Ditkin, V .A. and Prudnikov, A.P . ; Integral transforms and operational calculus . Pergamon Press, Oxford, London, Frankfurt, 1965","work_id":"ab6daeef-8055-46a2-b926-f0b0bb16086d","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"and Paris, R.B.; On integrals involving quotie nts of hyperbolic functions, Journal of the Ramanujan Mathematical Society, 36(1), 1-10, 2021","work_id":"293a8073-d788-411e-90bf-14cf4c1f243b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":20,"snapshot_sha256":"08ce955dad4272e3cbbbaf25267fd5195a81563df6859b24d0238c4a9ec4c736","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"9a3059b46eb030578978b60bf3766233b800e43f7abc0503b250cc93df0ce80b"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}