  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'.
