Finite-Part Integration of the Generalized Stieltjes Transform and its dominant asymptotic behavior for small values of the parameter. II. Non-integer orders

Abstract

The paper constitutes the second part on the subject of finite part integration of the generalized Stieltjes transform $S_{\lambda}[f]=\int_0^{\infty} f(x) (\omega+x)^{-\lambda}\mathrm{d}x$ about $\omega = 0$ where now $\lambda$ is a non-integer positive real number. Divergent integrals with singularities at the origin are induced by writing $(\omega+x)^{-\lambda}$ as a binomial expansion about $\omega = 0$ and interchanging the order of operations of integration and summation. The prescription of finite part integration is then implemented by interpreting these divergent integrals as finite part integrals which are rigorously represented as complex contour integrals. The same contour is then used to express $S_{\lambda}[f]$ itself as a complex contour integral. This led to the recovery of the terms missed by naive term-wise integration which themselves are finite parts of divergent integrals whose singularity is at the finite upper limit of integration. When the function $f(x)$ has a zero at the origin of order $m=0,1,\dots$ such that $m-\lambda<0$ the correction terms missed out by naive term by term integration give the dominant contribution to $S_{\lambda}[f]$ as $\omega\to 0$. Otherwise, the correction term is sub-dominant to the leading convergent terms in the naive term by term integration. We apply these results by obtaining exact and asymptotic representations of the Kummer and Gauss hypergeometric functions by evaluating their known Stieltjes integral representations. We then apply the method of finite part integration to obtain the asymptotic behavior of a generalization of the Stieltjes integral which is relevant in the calculation of the effective index of refraction of a shallow potential well.