Finite-part integration in the presence of competing singularities: Transformation equations for the hypergeometric functions arising from finite-part integration

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