Hadamard Regularization

TL;DR

This paper develops a mathematical framework using Hadamard regularization to handle singular functions with point-like singularities, crucial for high-order post-Newtonian approximations in general relativity.

Contribution

It introduces a systematic approach to define and manipulate pseudo-functions and derivatives for singular functions, extending classical distribution theory.

Findings

Defined Hadamard 'partie finie' for singular functions.

Constructed delta-pseudo-functions generalizing Dirac distributions.

Introduced a new derivative operator for pseudo-functions.

Abstract

Motivated by the problem of the dynamics of point-particles in high post-Newtonian (e.g. 3PN) approximations of general relativity, we consider a certain class of functions which are smooth except at some isolated points around which they admit a power-like singular expansion. We review the concepts of (i) Hadamard ``partie finie'' of such functions at the location of singular points, (ii) the partie finie of their divergent integral. We present and investigate different expressions, useful in applications, for the latter partie finie. To each singular function, we associate a partie-finie (Pf) pseudo-function. The multiplication of pseudo-functions is defined by the ordinary (pointwise) product. We construct a delta-pseudo-function on the class of singular functions, which reduces to the usual notion of Dirac distribution when applied on smooth functions with compact support. We introduce and analyse a new derivative operator acting on pseudo-functions, and generalizing, in this context, the Schwartz distributional derivative. This operator is uniquely defined up to an arbitrary numerical constant. Time derivatives and partial derivatives with respect to the singular points are also investigated. In the course of the paper, all the formulas needed in the application to the physical problem are derived.