Complex Tauberian theorems for the one-sided and two-sided Stieltjes transform

In the paper we prove one-sided and two-sided complex Tauberian theorems for the Stieltjes transform.
Tauberian theorems of the type of Keldysh theorem [1] with a remainder for the one-sided Stieltjes transform in the real domain were studied in [2]–[5]. Theorems of such type in the complex domain have not been studied in detail. Here only one result [6], [7] for the one-sided case is known.