  The paper surveys the basic properties of generalized Stieltjes functions
including some new ones. We introduce the notion of the exact Stieltjes order
and give a criterion of exactness, simple sufficient conditions and some
prototypical examples. The paper includes an appendix, where we define the left
sided Riemann-Liouville and the right sided Kober-Erdelyi fractional integrals
of measures supported on half axis and give inversion formulas for them.
