Finite-part integration is a recently introduced method of evaluating convergent integrals by means of the finite-part of divergent integrals [E. A. Galapon, Proc. R. Soc., A 473, 20160567 (2017)]. Current application of the method involves exact and asymptotic evaluation of the generalized Stieltjes transform ∫0af(x)/(ω+x)ρdx under the assumption that the extension of f(x) in the complex plane is entire. In this paper, the method is elaborated further and extended to accommodate the presence of competing singularities of the complex extension of f(x). Finite-part integration is then applied to derive consequences of known Stieltjes integral representations of the Gauss function and the generalized hypergeometric function that involve Stieltjes transforms of functions with complex extensions having singularities in the complex plane. Transformation equations for the Gauss function are obtained from which known transformation equations are shown to follow. In addition, building on the results for the Gauss function, transformation equations involving the generalized hypergeometric function 3F2 are derived.