Consider the class of functions which are smooth () on except for the two points and , around which they admit a power-like singular expansion of the typesingular coefficients of . If and belong to so does the ordinary product , as well as the ordinary gradient . We define the Hadamard “partie finie” of at the location of the singular point 1 as partie finie is “non-distributive” in the sense that in general. The second notion of Hadamard partie finie () concerns that of the integral , which is generically divergent at the location of the two singular points and (we assume no divergence at infinity). It is defined by partie-finie integral depends on two strictly positive constants and , associated with the logarithms present in Eq. (119). See Ref.  for alternative expressions of the partie-finie integral.
To any we associate the partie finie pseudo-function , namely a linear form acting on , which is defined by the duality bracket. The product of pseudo-functions coincides, by definition, with the ordinary pointwise product, namely . An interesting pseudo-function, constructed in Ref.  on the basis of the Riesz delta function , is the delta-pseudo-function , which plays a role analogous to the Dirac measure in distribution theory, . It is defined by partie finie of as given by Eq. (118). From the product of with any we obtain the new pseudo-function , that is such that partie finie, to replace within the pseudo-function by its regularized value: . The object has no equivalent in distribution theory.
Next, we treat the spatial derivative of a pseudo-function of the type , namely . Essentially, we require (in Ref. ) that the so-called rule of integration by parts holds. By this we mean that we are allowed to freely operate by parts any duality bracket, with the all-integrated (“surface”) terms always zero, as in the case of non-singular functions. This requirement is motivated by our will that a computation involving singular functions be as much as possible the same as a computation valid for regular functions. By definition,. Notably, we see that the integral of a gradient is always zero: . This should certainly be the case if we want to compute a quantity (e.g., a Hamiltonian density) which is defined only modulo a total divergence. We pose :  for the definition of another derivative operator, representing in fact the most general derivative satisfying the same properties as the one defined by Eq. (125) and, in addition, the commutation of successive derivatives (or Schwarz lemma)18. The distributional derivative (124, 125, 126) does not satisfy the Leibniz rule for the derivation of a product, in accordance with a general theorem of Schwartz . Rather, the investigation in Ref.  has suggested that, in order to construct a consistent theory (using the “ordinary” product for pseudo-functions), the Leibniz rule should in a sense be weakened, and replaced by the rule of integration by part (123), which is in fact nothing but an “integrated” version of the Leibniz rule.
The Hadamard regularization is defined by Eq. (118) in a preferred spatial hypersurface of a coordinate system, and consequently is not a priori compatible with the requirement of global Lorentz invariance. Thus we expect that the equations of motion in harmonic coordinates (which, we recall, manifestly preserve the global Lorentz invariance) should exhibit at some stage a violation of the Lorentz invariance due to the latter regularization. In fact this occurs exactly at the 3PN order. Up to the 2.5PN level, the use of the regularization is sufficient in order to get some Lorentz-invariant equations of motion . To deal with the problem at 3PN a Lorentz-invariant regularization, denoted , was introduced in Ref. . It consists of performing the Hadamard regularization within the spatial hypersurface that is geometrically orthogonal (in a Minkowskian sense) to the four-velocity of the particle. The regularization differs from the simpler regularization by relativistic corrections of order at least. See Ref.  for the formulas defining this regularization in the form of some infinite power series in the relativistic parameter . The regularization plays a crucial role in obtaining the equations of motion at the 3PN order in Refs. [21, 22].
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