International Journal of Mathematics and Mathematical Sciences
Volume 4 (1981), Issue 1, Pages 67-88
doi:10.1155/S0161171281000045

Abelian theorems for the Stieltjes transform of functions. II

Abstract

An initial (final) value Abelian theorem concerning transforms of functions is a result in which known behavior of the function as its domain variable approaches zero (approaches ∞) is used to infer the behavior of the transform as its domain variable approaches zero (approaches ∞). We obtain such theorems in this paper concerning the Stieltjes transform. In our results all parameters are complex; the variable s of the transform is complex in the right half plane; and the initial (final) value Abelian theorems are obtained as |s|→0(|s|→∞) within an arbitrary wedge in the right half plane.