Postprint version. Published in Journal of Approximation Theory, Volume 121, January 1, 2003, pages 54-60. Copyright © 2003 Elsevier. The definitive version is available at http://dx.doi.org/10.1016/S0021-9045(02)00042-4.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
The pointwise limit S of a sequence of Stieltjes transforms (Sn) of real Borel probability measures (Pn) is itself the Stieltjes transform of a Borel p.m. P if and only if iy S(iy) →−1as y →∞, in which case Pn converges to P in distribution. Applications are given to several problems in mathematical physics.