论文标题
在竞争奇异性的情况下，有限零件集成：由有限零件集成产生的超几何函数的转换方程
Finite-Part Integration in the Presence of Competing Singularities: Transformation Equations for the hypergeometric functions arising from Finite-Part Integration
论文作者
论文摘要
有限零件集成是最近引入的方法，是通过发散积分的有限部分评估收敛积分的方法[E.A.加拉彭，{\ it proc。 R. Soc。 A 473，20160567}（2017）]。该方法的当前应用涉及广义stieltjes的精确和渐近评估，$ \ int_0^a f（x）/（ω+ x）^ρ\，\ mathrm {d} x $在复杂平面中$ f（x）$的扩展为整体。在本文中，该方法被进一步详细介绍并扩展，以适应$ f（x）$的复杂扩展的竞争奇异性。然后将有限部分积分应用于导致高斯函数的已知stieltjes积分表示的后果和广义的超几何函数，该功能涉及功能的变换，复杂的扩展具有复杂平面的奇异性。高斯函数的转换方程是从已知的转换方程中获得的。同样，构建了高斯函数的结果，涉及涉及广义超几何函数$ \，_ 3f_2 $的转换方程。
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 $\int_0^a f(x)/(ω+ x)^ρ \, \mathrm{d}x$ 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.
