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Hadamard Regularization

Luc Blanchet, Guillaume Faye

TL;DR

The paper develops Hadamard regularization (partie finie) for functions with two point singularities, providing a coherent framework of pseudo-functions ${\rm Pf}F$ and a derivative operator that extends distribution theory to singular fields relevant for high-order post-Newtonian gravity. It systematically constructs and analyzes the partie finie of divergent integrals, gradients, Poisson-type integrals, and delta-like objects, and introduces a derivative ${\sc D}_i[F]$ with an accompanying homogeneous part to ensure commutativity of higher-order derivatives. The work also connects Hadamard regularization to analytic continuation, angular integration, and Riesz delta-functions, and develops a robust calculus for time-dependent singularities to model point-particle dynamics at 3PN order. These tools enable precise, regulator-dependent computations in the two-body GR problem, with direct implications for gravitational-wave physics and high-precision celestial mechanics.

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.

Hadamard Regularization

TL;DR

The paper develops Hadamard regularization (partie finie) for functions with two point singularities, providing a coherent framework of pseudo-functions and a derivative operator that extends distribution theory to singular fields relevant for high-order post-Newtonian gravity. It systematically constructs and analyzes the partie finie of divergent integrals, gradients, Poisson-type integrals, and delta-like objects, and introduces a derivative with an accompanying homogeneous part to ensure commutativity of higher-order derivatives. The work also connects Hadamard regularization to analytic continuation, angular integration, and Riesz delta-functions, and develops a robust calculus for time-dependent singularities to model point-particle dynamics at 3PN order. These tools enable precise, regulator-dependent computations in the two-body GR problem, with direct implications for gravitational-wave physics and high-precision celestial mechanics.

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.

Paper Structure

This paper contains 24 sections, 16 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Hadamard partie finie
  3. A class of singular functions
  4. Partie finie of a singular function
  5. Partie-finie integrals
  6. The partie finie of a divergent integral
  7. Partie-finie integral of a gradient
  8. Parties finies and the Riesz delta-function
  9. Alternative forms of the partie finie
  10. Partie finie based on analytic continuation
  11. Partie finie based on angular integration
  12. Partie finie of Poisson integrals
  13. Partie finie pseudo-functions
  14. A class of pseudo-functions
  15. A Dirac delta-pseudo-function
  16. ...and 9 more sections

Key Result

Lemma 1

The partie finie at 1 of the gradient $\partial_iF$ (as defined outside the singularities) of any function $F\in {\cal F}$ satisfies

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Definition 4
  • Theorem 1
  • Lemma 2
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • ...and 17 more