2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/101293We define a period pairing for flat, irregular singular, rank one connections, satisfying a technical condition regarding its stationary set, on complex surfaces between de Rham cohomology of the connection and a modified singular homology, the rapid decay homology. We prove that this gives a perfect duality. A 2-dimensional version of confluent hypergeometrics are contructed as an example.28 pages; previous version partly incorrect (for higher rank case), new version contains case of line bundles and a new example. v3: technical assumption inserted (Definition 1.1)Algebraic GeometryNumber Theory14F40; 32S40; 25A20; 32C38text