A Tauberian theorem for laplace transforms with pseudofunction boundary Behavior

The 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'.