The Cauchy principal value and the Hadamard finite part integral as values of absolutely convergent integrals

The divergent integral ∫ a b f ( x ) ( x − x 0 ) − n − 1 d x , for −∞ < a < x 0 < b < ∞ and n = 0, 1, 2, …, is assigned, under certain conditions, the value equal to the simple average of the contour integrals ∫ C ± f(z)(z − x 0)−n−1dz, where C + (C −) is a path that starts from a and en...

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Bibliographic Details
Published in:Journal of mathematical physics 2016-03, Vol.57 (3), p.1
Main Author: Galapon, Eric A.
Format: Article
Language:English
Subjects:
Asymptotic methods
Convergence
Integrals
Mathematical analysis
Numerical analysis
Physics
Theorems
Treasury bonds
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Summary:The divergent integral ∫ a b f ( x ) ( x − x 0 ) − n − 1 d x , for −∞ < a < x 0 < b < ∞ and n = 0, 1, 2, …, is assigned, under certain conditions, the value equal to the simple average of the contour integrals ∫ C ± f(z)(z − x 0)−n−1dz, where C + (C −) is a path that starts from a and ends at b and which passes above (below) the pole at x 0. It is shown that this value, which we refer to as the analytic principal value, is equal to the Cauchy principal value for n = 0 and to the Hadamard finite-part of the divergent integral for positive integer n. This implies that, where the conditions apply, the Cauchy principal value and the Hadamard finite-part integral are in fact values of absolutely convergent integrals. Moreover, it leads to the replacement of the boundary values in the Sokhotski-Plemelj-Fox theorem with integrals along some arbitrary paths. The utility of the analytic principal value in the numerical, analytical, and asymptotic evaluations of the principal value and the finite-part integral is discussed and demonstrated.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.4943300