= Contour damping for bounded Laplace transforms
{title2=$e^{Tz}(1+z^2/R^2)/z$}
The displayed kernel has residue one at zero and vanishes at the endpoints of a radius-$R$ imaginary-axis segment. On either radius-$R$ semicircle, $|1+z^2/R^2|=2|\operatorname{Re}z|/R$, cancelling the inverse real-part bound on a bounded function's <Laplace transform> tail. Use the full transform only on a right semicircle and a thin analytic leftward path; deform the finite-interval entire transform to the large left semicircle. This proves the <Newman Tauberian theorem> without assuming analytic continuation across an entire left half-disk.
Back to article page
