2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69467The prime number theorem provided the chief impulse for complex Tauberian theory, in which the boundary behavior of a transform in the complex plane plays a crucial role. We consider Laplace transforms of bounded functions. Our Tauberian theorem does not allow first-order poles on the imaginary axis, but any milder singularities, characterized by pseudofunction boundary behavior, are permissible. In this context we obtain a useful Tauberian theorem by exploiting Newman's `contour method'.11 pages, 1 figureClassical Analysis and ODEsComplex VariablesPrimary: 40E05; Secondary: 11M45, 11N05, 42A38, 44A10, 46F20text