Finite-Part Integration in the Presence of Competing Singularities: Transformation Equations for the hypergeometric functions arising from Finite-Part Integration

TL;DR

This paper extends finite-part integration to handle functions with complex singularities, deriving new transformation equations for hypergeometric functions from Stieltjes transform representations.

Contribution

It advances finite-part integration methodology to include competing singularities, leading to new transformation formulas for hypergeometric functions.

Findings

Derived new transformation equations for the Gauss hypergeometric function.

Extended finite-part integration to functions with complex singularities.

Connected transformation formulas to known hypergeometric identities.

Abstract

Finite-part integration is a recently introduced method of evaluating convergent integrals by means of the finite part of divergent integrals [E.A. Galapon, {\it Proc. R. Soc. A 473, 20160567} (2017)]. Current application of the method involves exact and asymptotic evaluation of the generalized Stieltjes transform $\int_0^a f(x)/(\omega + x)^{\rho} \, \mathrm{d}x$ 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 which 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. Also, building on the results for the Gauss function, transformation equations involving the generalized hypergeometric function $\,_3F_2$ are derived.