Complex inversion formula for the distributional Stieltjes transform

Abstract

There are several approaches to the Stieltjes transform of generalized functions ([1, 10, 5, 6, 3, 2]). In this paper we use the definition of the distributional Stieltjes transform of index ρ (ρ ∈ ℝ\(−ℕ0); ℕ0 = ℕ∪{0}), Sρ-transform, given by Lavoine and Misra [3]. The Sρ-transform is defined for a subspace of the Schwartz space (ℝ) while in [10, 5, 6, 2] the Stieltjes transform is defined for the elements of appropriate spaces of generalized functions. In these spaces differentiation is not defined which means that the Stieltjes transform of some important distributions, for example δ(k)(x − a), a≧0, k ∈ ℕ, is meaningless in the sense of [10, 5, 6, 2]. It is easy to see that the distributions δ(k)(x − a), a≧0, k ∈ ℕ, have the Sρ-transform for ρ>−k, ρ∈ℝ\(−ℕ0). These facts favour the approach to the Stieltjes transform given in [3].