Affiliation:
1. Theoretical Physics Group, National Institute of Physics, University of the Philippines, Diliman Quezon City, 1101 Philippines
Abstract
Finite-part integration is a recent method of evaluating a convergent integral in terms of the finite-parts of divergent integrals deliberately induced from the convergent integral itself [E. A. Galapon, The problem of missing terms in term by term integration involving divergent integrals, Proc. R. Soc. A 473 (2017) 20160567]. Within the context of finite-part integration of the Stieltjes transform of functions with logarithmic growths at the origin, the relationship is established between the analytic continuation of the Mellin transform and the finite-part of the resulting divergent integral when the Mellin integral is extended beyond its strip of analyticity. It is settled that the analytic continuation and the finite-part integral coincide at the regular points of the analytic continuation. To establish the connection between the two at the isolated singularities of the analytic continuation, the concept of regularized limit is introduced to replace the usual concept of limit due to Cauchy when the later leads to a division by zero. It is then shown that the regularized limit of the analytic continuation at its isolated singularities equals the finite-part integrals at the singularities themselves. The treatment gives the exact evaluation of the Stieltjes transform in terms of finite-part integrals and yields the dominant asymptotic behavior of the transform for arbitrarily small values of the parameter in the presence of arbitrary logarithmic singularities at the origin.
Funder
UP-System Enhanced Creative Work and Research
Publisher
World Scientific Pub Co Pte Ltd
Subject
Applied Mathematics,Analysis
Cited by
1 articles.
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1. Continuation of the Stieltjes series to the large regime by finite-part integration;Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences;2023-08