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Hadamard's Finite Part
Author(s) -
Jones D. S.
Publication year - 1996
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(19960910)19:13<1017::aid-mma723>3.0.co;2-2
Subject(s) - mathematics , hadamard transform , infinity , gravitational singularity , convolution (computer science) , logarithm , algebraic number , range (aeronautics) , mathematical analysis , multiplication (music) , pure mathematics , algebra over a field , combinatorics , materials science , machine learning , artificial neural network , computer science , composite material
A systematic way of ascribing values to integrals which diverge at infinity is described. For integrals with algebraic or logarithmic behaviour it is akin to Hadamard's finite part for integrands which have singularities at finite points. The power of the method is illustrated by examples of particular integrals and the convolution of distributions which are beyond the range of standard theory. Both one‐dimensional and multi‐dimensional integrals are included. The theory also enables the definition of products of distributions which are considered normally to be too singular for multiplication.