Laurent Baratchart

Sylvain Chevillard

Juliette Leblond

Eduardo Andrade Lima

Dmitry Ponomarev

Eduardo Andrade Lima

Asymptotic method for estimating magnetic moments from field measurements on a planar grid

Introduction

Estimating the net moment of a magnetization distribution is a fundamental issue in studies of magnetized materials conducted in research areas ranging from geosciences and material sciences to medical imaging. Standard magnetometers infer the net moment from a set of measurements of the magnetic field taken at a fixed distance from the sample or integrated over a defined sensing area/volume. Typically, assumptions are made about the shape of the specimen under analysis and the uniformity of its magnetization so that simple and explicit formulas connect the magnetic dipole moment with the measured magnetic field or flux [START_REF] Collinson | Methods in rock magnetism and paleomagnetism: techniques and instrumentations[END_REF].

While such an approximation is valid under certain conditions (e.g., the field is measured with sufficient accuracy far enough from the support of the magnetization; the magnetization is uniform and its support nearly spherical; the support of the magnetization is much smaller than the sensing region), it is not satisfactory when dealing with very weakly magnetized objects whose fields may get easily mixed with those of spurious magnetic sources away from the sample. Still, analyzing weak magnetization distributions is of considerable interest, e.g., in paleomagnetism and biomagnetism. To this effect, scanning magnetic microscopes capable of measuring very weak fields at submillimetric distances from an object have been developed in recent years. These advances in instrumentation have driven the need for developing alternative techniques for estimating the net moment of a magnetization distribution from a set of magnetic field measurements.

From a mathematical viewpoint, the problem consists of recovering the mean of a compactly supported vector field from knowledge of the gradient of the potential of its divergence in some region near the support. Indeed, it follows from Maxwell's equations for the magnetostatic case [START_REF] Jackson | Classical Electrodynamics[END_REF]Ch. 5] that the magnetization distribution and its associated magnetic field are connected through an elliptic partial differential equation of Poisson type. More precisely, the magnetic field is the gradient of a scalar magnetic potential whose Laplacian is the divergence of the magnetization.

The main feature of this inverse problem is the geometry of the measurement set. In this work, we consider the case where measurements are taken on a plane which does not intersect the support of the magnetization. For instance, this setup is typical of experiments conducted with scanning superconducting quantum interference device (SQUID) microscopes when studying rock samples, see [START_REF] Weiss | Paleomagnetic analysis using SQUID microscopy[END_REF][START_REF] Kirtley | Scanning SQUID microscopy[END_REF]. In practice, two additional constraints complicate the situation further. The first is that only a single component of the field can be measured, namely the one that is orthogonal to the measurement plane: owing to the proximity to the sample and the very high sensitivity required, it is very difficult to accommodate more than one SQUID sensor without compromising accuracy, magnetic moment sensitivity, and spatial resolution. The second constraint is that measurements can only be performed on a finite portion of the measurement plane, close enough to the sample.

In analogy with the expansion of a dipolar field at infinity, the first and most basic issue comprises obtaining formulas connecting the normal component of the magnetic field on a plane to the integral of the magnetization (i.e., net moment) that generates it, in such a way that knowledge of this component on part of that plane yields approximate formulas for the moment. Surprisingly, no such formulas seem readily available in the literature and the goal of the present paper is to provide useful expressions for estimating the net moment of compactly supported magnetizations from measurements of the normal component of its associated magnetic field taken on a portion of a plane.

We point out that net moment estimation is part of the larger inverse problem of full magnetization recovery from magnetic field measurements. The latter is ill-posed, not even injective. For thin supports (that can be identified with planar sets), non-uniqueness issues are analyzed in [START_REF] Baratchart | Characterizing kernels of operators related to thin-plate magnetizations via generalizations of hodge decompositions[END_REF] and some recovery schemes are considered in [START_REF] Lima | Fast inversion of magnetic field maps of unidirectional planar geological magnetization[END_REF] for unidirectional magnetizations. In contrast, net moment recovery is well-posed in that magnetizations producing the same normal component for the magnetic field on an open set of a plane must have the same moment (see [START_REF] Baratchart | Silent and equivalent magnetic distributions on thin plates[END_REF]). Besides, the net moment provides valuable information on the magnetization itself, which may be used in full recovery schemes. This further motivates investigating recovery schemes for the moment of the magnetization.

Up to a rotation, we may assume for the ease of discussion that the measurement plane is horizontal and that the magnetization distribution is located below this plane. The questions we face are thus: (i) how can the vertical component of the magnetic field on a portion of the horizontal plane be used to yield an approximation of the moment of the magnetization generating that field? (ii) How does the error decay when that portion of horizontal plane grows large? In this regard, we mention that such asymptotics for the net moment were taken up in [START_REF] Ponomarev | Some inverse problems with partial data[END_REF]Part III,Sec. 5,[START_REF] Kirtley | Scanning SQUID microscopy[END_REF] for circular measurement areas using Fourier techniques and tools from harmonic analysis. As we will see below, formulas of a similar type can be obtained using elementary properties of homogeneous polynomials and Taylor expansions for rectangular measurement sets.

In the present work, we carry out in detail the corresponding computations when the measurement set is a square. The overview is as follows. The problem is set up in Section 2 and the main approximation results are stated in Section 3. Their proofs are given in Sections 4, 5 and in the Appendix. We discuss in Section 6 how these results can be used to provide estimates of the net moment of a magnetization when measurements of the field 𝐵 3 it generates are available. Finally, Section 7 contains concluding remarks.

Notation and problem setting

Given 𝑠 , 𝑟 > 0, we consider a parallelepiped 𝒜 = [-𝑠, 𝑠] 2 × [0, 𝑟] ⊂ R 3 to contain the volume of the sample. Arbitrary points of R 3 will be denoted as ⃗ 𝑥 = (𝑥 1 , 𝑥 2 , 𝑥 3 ), while ⃗ 𝑡 = (𝑡 1 , 𝑡 2 , 𝑡 3 ) will represent an arbitrary point of 𝒜. For 𝑖 = 1, 2, 3, we suppose we are given a real-valued function 𝑚 𝑖 ∈ 𝐿 1 (𝒜), the Lebesgue space of summable functions on 𝒜. We denote by ⃗ 𝑚 the magnetization vector field (𝑚 1 , 𝑚 2 , 𝑚 3 ) of components 𝑚 𝑖 . A volumetric magnetization compactly supported on the slab 𝒜 is modeled by the vector field on R 3 , ⃗ 𝑥 ↦ → ̃ ⃗ 𝑚(⃗ 𝑥) where, for 𝑖 = 1, 2, 3, ̃ 𝑚 𝑖 denotes the function 𝑚 𝑖 extended by 0 outside 𝒜, i.e., ̃ 𝑚 𝑖 (⃗ 𝑥) = 𝑚 𝑖 (⃗ 𝑥) if ⃗ 𝑥 ∈ 𝒜 and ̃ 𝑚 𝑖 (⃗ 𝑥) = 0 otherwise. For any 𝑚 ∈ 𝐿 1 (𝒜), we denote by ⟨𝑚⟩ the net moment given by the mean value of 𝑚:

⟨𝑚⟩ = ∫∫∫ 𝒜 𝑚( ⃗ 𝑡) d ⃗ 𝑡.
More generally, the net moment of 𝑚 is its 0-th order moment, while the 1-st order moments are the quantities ⟨𝑡 1 𝑚⟩, ⟨𝑡 2 𝑚⟩, and ⟨𝑡 3 𝑚⟩, the 2-nd order moments are the quantities ⟨𝑡 𝑖 𝑡 𝑗 𝑚⟩ (with arbitrary 𝑖 and 𝑗 in {1, 2, 3}), etc.

As recalled in [START_REF] Baratchart | Characterizing kernels of operators related to thin-plate magnetizations via generalizations of hodge decompositions[END_REF], the magnetic field produced by the magnetized slab (𝒜, ⃗ 𝑚) is B = -𝜇 0 ∇𝜑, where 𝜇 0 = 4𝜋 × 10 -7 and 𝜑 is the scalar magnetic potential defined at each point ⃗ 𝑥 ̸ ∈ 𝒜 by

𝜑(⃗ 𝑥) = 1 4𝜋 ∫∫∫ 𝒜 ⟨ ⃗ 𝑚( ⃗ 𝑡), ⃗ 𝑥 -⃗ 𝑡⟩ ‖⃗ 𝑥 -⃗ 𝑡‖ 3 d ⃗ 𝑡 , (1)
with ‖⃗ 𝑥‖ = √ 𝑥 ‖⃗ 𝑥‖ 3 , for 𝑖 = 1, 2, 3, and by ⋆ the convolution product between functions defined on R 3 , we can express Equation (1) as the following shorter expression:

𝜑(⃗ 𝑥) = 1 4𝜋 (𝑃 1 ⋆ ̃ 𝑚 1 + 𝑃 2 ⋆ ̃ 𝑚 2 + 𝑃 3 ⋆ ̃ 𝑚 3 ) (⃗ 𝑥). ( 2
)
In the following, we will assume that we have measurements of the vertical component 𝐵 3 of B at a given height 𝑥 3 = 𝑧 > 𝑟. This defines a function on the plane and we denote by

𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) = 𝐵 3 (𝑥 1 , 𝑥 2 , 𝑧) its value at a point (𝑥 1 , 𝑥 2 ) ∈ R 2 .
For ⃗ 𝑥 ̸ ∈ 𝒜, we have:

- 4𝜋 𝜇 0 𝐵 3 [ ⃗ 𝑚, 𝑥 3 ](𝑥 1 , 𝑥 2 ) = 𝜕 𝑥 1 (𝑃 3 ⋆ ̃ 𝑚 1 -𝑃 1 ⋆ ̃ 𝑚 3 )(⃗ 𝑥) + 𝜕 𝑥 2 (𝑃 3 ⋆ ̃ 𝑚 2 -𝑃 2 ⋆ ̃ 𝑚 3 )(⃗ 𝑥) . ( 3
)
This is easily checked by a direct computation: on the one hand, starting from

𝐵 3 [ ⃗ 𝑚, 𝑥 3 ](𝑥 1 , 𝑥 2 ) = -𝜇 0 𝜕 𝑥 3 𝜑(⃗ 𝑥)
and computing the derivative with respect to 𝑥 3 of 𝜑 as given in Equation (1); on the other hand, explicitly computing the derivatives with respect to 𝑥 1 and 𝑥 2 in Equation (3). Another (deeper but more involved) way of seeing it consists in recognizing Poisson and Riesz transforms in Equation (2) and using their properties with respect to differentiation (see [START_REF] Baratchart | Silent and equivalent magnetic distributions on thin plates[END_REF] for instance, where this is done in the case of a 2D slab).

Finally, for 𝑅 > 0, we introduce the planar measurement areas 2. The symmetry of these measurement areas with respect to the origin is of essential use in the computations.

𝑄 𝑅 = [-𝑅, 𝑅] 2 (square), 𝑆 𝑅 = {(𝑥 1 , 𝑥 2 ) ∈ R 2 , |𝑥 1 | + |𝑥 2 | ≤ 𝑅} (diamond) and 𝐴 𝑅 = {(𝑥 1 , 𝑥 2 ) ∈ R 2 , 𝑥 2 1 + 𝑥 2 2 ≤ 𝑅 2 } (disk) as illustrated in Figure

Main results

Our main result is summed up in the following theorem that provides asymptotic expansions (as 𝑅 goes large) of simple integrals involving 𝐵 3 [ ⃗ 𝑚, 𝑧], in terms of the successive moments of the magnetization ⃗ 𝑚.

Theorem 1. Let notation and assumptions be as in Section 2. On the square 𝑄 𝑅 , it holds that:

∫∫ 𝑄 𝑅 𝑥 1 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 𝜇 0 2 ⟨𝑚 1 ⟩ + 3𝜇 0 𝜋𝑅 √ 2 (⟨𝑡 1 𝑚 3 ⟩ + ⟨𝑡 3 𝑚 1 ⟩ -𝑧⟨𝑚 1 ⟩) + 𝒪 ( 1 𝑅 3 ) , (4)
∫∫ 𝑄 𝑅 𝑥 2 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 𝜇 0 2 ⟨𝑚 2 ⟩ + 3𝜇 0 𝜋𝑅 √ 2 (⟨𝑡 2 𝑚 3 ⟩ + ⟨𝑡 3 𝑚 2 ⟩ -𝑧⟨𝑚 2 ⟩) + 𝒪 ( 1 𝑅 3 ) , ( 5
)
∫∫ 𝑄 𝑅 𝑅 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 2𝜇 0 𝜋 √ 2 ⟨𝑚 3 ⟩ + 5𝜇 0 4𝜋𝑅 2 √ 2 ( -2𝑧 2 ⟨𝑚 3 ⟩ + 2𝑧 (2⟨𝑡 3 𝑚 3 ⟩ -⟨𝑡 1 𝑚 1 ⟩ -⟨𝑡 2 𝑚 2 ⟩) + ⟨𝑡 2 1 𝑚 3 ⟩ + ⟨𝑡 2 2 𝑚 3 ⟩ -2⟨𝑡 2 3 𝑚 3 ⟩ + 2⟨𝑡 1 𝑡 3 𝑚 1 ⟩ + 2⟨𝑡 2 𝑡 3 𝑚 2 ⟩ ) + 𝒪 ( 1 𝑅 3 ) , ( 6
)
On the diamond 𝑆 𝑅 , it holds that:

∫∫ 𝑆 𝑅 𝑥 1 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 𝜇 0 2 ⟨𝑚 1 ⟩ + 3𝜇 0 𝜋𝑅 (⟨𝑡 1 𝑚 3 ⟩ + ⟨𝑡 3 𝑚 1 ⟩ -𝑧⟨𝑚 1 ⟩) + 𝒪 ( 1 𝑅 3 ) , ( 7
)
∫∫ 𝑆 𝑅 𝑥 2 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 𝜇 0 2 ⟨𝑚 2 ⟩ + 3𝜇 0 𝜋𝑅 (⟨𝑡 2 𝑚 3 ⟩ + ⟨𝑡 3 𝑚 2 ⟩ -𝑧⟨𝑚 2 ⟩) + 𝒪 ( 1 𝑅 3 ) , ( 8
)
∫∫ 𝑆 𝑅 𝑅 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 2𝜇 0 𝜋 ⟨𝑚 3 ⟩ + 5𝜇 0 2𝜋𝑅 2 ( -2𝑧 2 ⟨𝑚 3 ⟩ + 2𝑧 (2⟨𝑡 3 𝑚 3 ⟩ -⟨𝑡 1 𝑚 1 ⟩ -⟨𝑡 2 𝑚 2 ⟩) + ⟨𝑡 2 1 𝑚 3 ⟩ + ⟨𝑡 2 2 𝑚 3 ⟩ -2⟨𝑡 2 3 𝑚 3 ⟩ + 2⟨𝑡 1 𝑡 3 𝑚 1 ⟩ + 2⟨𝑡 2 𝑡 3 𝑚 2 ⟩ ) + 𝒪 ( 1 𝑅 3 ) . ( 9
)
Remark: we do not prove it in this article (for the sake of keeping computations reasonably simple) but the 𝒪(1/𝑅 3 ) term of Equations ( 6) and ( 9) is actually a 𝒪(1/𝑅 4 ).

The forthcoming Section 4 will establish Equations ( 4) and [START_REF] Kirtley | Scanning SQUID microscopy[END_REF]. We shall deduce all the other equations from the latter by means of appropriate changes of variable. This is performed in the dedicated Section 5. We conclude the present section with some comments about Theorem 1.

Thin slab hypothesis.

In some contexts, the thickness of the magnetization slab is much smaller than its distance to the measurement plane, such that, from a practical point of view, one can consider its height 𝑟 as being 0, i.e., the slab is embedded in a plane instead of a volume. The magnetization is then modeled as a planar magnetization distribution [START_REF] Baratchart | Characterizing kernels of operators related to thin-plate magnetizations via generalizations of hodge decompositions[END_REF]. If 𝑚 denotes a component of such a magnetization (which therefore depends only on variable 𝑡 1 and 𝑡 2 since in this case 𝒜 is simply the square [-𝑠, 𝑠] 2 ), its net moment is then given by ⟨𝑚⟩ = ∫∫ 𝒜 𝑚(𝑡 1 , 𝑡 2 ) d𝑡 1 d𝑡 2 . Higher order moments are defined accordingly, and when they involve 𝑡 3 they are simply equal to 0. In practice, asymptotic formulas for this thin-slab context are hence simply the same as those given in Theorem 1, when ignoring the higher-order moments containing 𝑡 3 .

Adjoint operator. Notice that, if 𝑢 is a function in 𝐿 1 (R 2 ), we have (for

𝑥 3 > 𝑟) ∫∫ R 2 𝑢(𝑥 1 , 𝑥 2 )𝐵 3 [ ⃗ 𝑚, 𝑥 3 ](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = ∫∫∫ 𝒜 𝑚 1 ( ⃗ 𝑡) 𝑣 1 ( ⃗ 𝑡) + 𝑚 2 ( ⃗ 𝑡) 𝑣 2 ( ⃗ 𝑡) + 𝑚 3 ( ⃗ 𝑡) 𝑣 3 ( ⃗ 𝑡) d ⃗ 𝑡 where ⃗ 𝑣( ⃗ 𝑡) = (𝑣 1 ( ⃗ 𝑡), 𝑣 2 ( ⃗ 𝑡), 𝑣 3 ( ⃗ 𝑡)) is given by ⃗ 𝑣( ⃗ 𝑡) = 𝜇 0 4𝜋 𝜕 𝑥 3 ∫∫ R 2 ⃗ 𝑡 -⃗ 𝑥 ‖ ⃗ 𝑡 -⃗ 𝑥‖ 3 𝑢(𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 . ( 10
)
This simply follows from the definition of B from 𝜑 and from Equation (1), using Fubini's theorem and the fact the derivation with respect to 𝑥 3 commutes with all involved integrals. One can also easily check (explicitly computing the derivatives) that

⃗ 𝑣( ⃗ 𝑡) = - 𝜇 0 4𝜋 ∇ ( (𝑡 3 -𝑥 3 ) ∫∫ R 2 𝑢(𝑥 1 , 𝑥 2 ) ‖ ⃗ 𝑡 -⃗ 𝑥‖ 3 d𝑥 1 d𝑥 2 ) (11)
where ∇ denotes the gradient: ∇ = (𝜕 𝑡 1 , 𝜕 𝑡 2 , 𝜕 𝑡 3 ). Equations [START_REF] Weiss | Paleomagnetic analysis using SQUID microscopy[END_REF] and (11) are reminiscent of formulas of the adjoint operator presented in [START_REF] Baratchart | Silent and equivalent magnetic distributions on thin plates[END_REF] in the case of magnetizations in 𝐿 2 (𝒜) where 𝒜 is a two-dimensional thin slab. Our proof of Equations ( 4) and ( 6) essentially consists in taking 𝑢(𝑥 1 , 𝑥 2 ) = 𝑥 1 𝜒 𝑄 𝑅 and 𝑢(𝑥 1 , 𝑥 2 ) = 𝑅𝜒 𝑄 𝑅 where 𝜒 𝑄 𝑅 denotes the characteristic function of the square 𝑄 𝑅 , and doing an asymptotic expansion of the corresponding function ⃗ 𝑣( ⃗ 𝑡) with respect to 𝑅 (and with 𝑥 3 being fixed to 𝑧). It turns out that Equation (11) can be explicitly computed in these cases (thanks to functions 𝑘 and ℓ introduced in Definition 1, see Proposition 1 and following in the next section). The main issue is then to ensure that the remainder of the asymptotic expansion commutes with the integration on ⃗ 𝑡 ∈ 𝒜, which we achieve by computing the explicit dependence of the remainder with respect to variable ⃗ 𝑡. As will appear in the next section, we actually proceed slightly differently mainly to keep computations reasonably simple (the explicit expression of ⃗ 𝑣( ⃗ 𝑡) obtained from Equation (11) is quite large, with many similar terms, so we believe it is easier to conduct the computations in a more ad hoc way), but the overall strategy follows the same idea.

Magnetic moment magnetometer.

Observe that, at first order, the integrals of Theorem 1 provide us with estimates for the quantities ⟨𝑚 𝑖 ⟩ (𝑖 = 1, 2, 3) and can hence be used to implement a moment magnetometer. This is in fact what motivated this work and this idea will be developed further in Section 6, where we combine the expansions on the square 𝑄 𝑅 and the diamond 𝑆 𝑅 to eliminate the term in 1/𝑅 in the expansions and achieve a better accuracy. Also,we point out the fact that, even though the remainders are given in Theorem 1 in the 𝒪(•) form (which is the usual notation for truncated asymptotic expansions) our proof actually computes bounds that are explicit with respect to the characteristic quantities 𝑠, 𝑧 and 𝑅 (namely the bounds of function 𝛿 18 , resp. 𝛿 23 , at the end of Section 4.2, resp. 4.3).

Disk geometry.

Similar asymptotic formulas can be obtained for the disk 𝐴 𝑅 centered at 0 and of radius 𝑅, namely it holds that:

∫∫ 𝐴 𝑅 𝑥 1 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 𝜇 0 2 ⟨𝑚 1 ⟩ + 3𝜇 0 4𝑅 (⟨𝑡 1 𝑚 3 ⟩ + ⟨𝑡 3 𝑚 1 ⟩ -𝑧⟨𝑚 1 ⟩) + 𝒪 ( 1 𝑅 3 ) , ( 12
)
∫∫ 𝐴 𝑅 𝑥 2 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 𝜇 0 2 ⟨𝑚 2 ⟩ + 3𝜇 0 4𝑅 (⟨𝑡 2 𝑚 3 ⟩ + ⟨𝑡 3 𝑚 2 ⟩ -𝑧⟨𝑚 2 ⟩) + 𝒪 ( 1 𝑅 3 ) , ( 13
) ∫∫ 𝐴 𝑅 𝑅 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 𝜇 0 2 ⟨𝑚 3 ⟩ + 3𝜇 0 8𝑅 2 ( -2𝑧 2 ⟨𝑚 3 ⟩ + 2𝑧 (2⟨𝑡 3 𝑚 3 ⟩ -⟨𝑡 1 𝑚 1 ⟩ -⟨𝑡 2 𝑚 2 ⟩) + ⟨𝑡 2 1 𝑚 3 ⟩ + ⟨𝑡 2 2 𝑚 3 ⟩ -2⟨𝑡 2 3 𝑚 3 ⟩ + 2⟨𝑡 1 𝑡 3 𝑚 1 ⟩ + 2⟨𝑡 2 𝑡 3 𝑚 2 ⟩ ) + 𝒪 ( 1 𝑅 4 ) . ( 14
)
We will not prove these formulas. They can be obtained from [9, Part III, Sec. 4 Proofs of Equations ( 4) and (6)

Preliminary results

Before going to the actual proof of Equations ( 4) and ( 6) we define functions and establish results that will be of constant use in what follows.

Definition 1. We define the R-valued functions 𝑓 , 𝑔, 𝑘 and ℓ for ⃗ 𝑥 ∈ R 3 with 𝑥 3 > 0 by:

𝑓 (⃗ 𝑥) = 𝑥 2 𝑥 2 1 + 𝑥 2 3 • 1 ‖⃗ 𝑥‖ , 𝑔(⃗ 𝑥) = -1 ‖⃗ 𝑥‖ , 𝑘(⃗ 𝑥) = 1 𝑥 3 arctan ( 𝑥 1 𝑥 2 𝑥 3 ‖⃗ 𝑥‖ ) , ℓ(⃗ 𝑥) = -arcsinh ( 𝑥 2 (𝑥 2 1 + 𝑥 2 3 ) 1/2 )
.

These functions are indefinite integrals of expressions that will naturally come up when rewriting the left hand sides of Equations ( 4) and ( 6). This is capsulized in the following proposition whose proof reduces to straightforward computations. Below, the symbol 𝜕 𝑥 𝑖 stands for the partial derivative with respect to the coordinate

𝑥 𝑖 (𝑖 = 1, 2, 3) while 𝜕 2 𝑥 1 𝑥 2 = 𝜕 𝑥 1 𝜕 𝑥 2 .
Proposition 1. For any ⃗ 𝑥 ∈ R 3 with 𝑥 3 > 0, we have

𝜕 𝑥 2 𝑓 (⃗ 𝑥) = 1/‖⃗ 𝑥‖ 3 , 𝜕 𝑥 1 𝑔(⃗ 𝑥) = 𝑥 1 /‖⃗ 𝑥‖ 3 , 𝜕 2 𝑥 1 𝑥 2 𝑘(⃗ 𝑥) = 1/‖⃗ 𝑥‖ 3 , 𝜕 2 𝑥 1 𝑥 2 ℓ(⃗ 𝑥) = 𝑥 1 /‖⃗ 𝑥‖ 3 .
We will need asymptotic expansions of expressions of the form 𝑓 (𝑅 -𝑡 1 , 𝑅 -𝑡 2 , 𝑧 -𝑡 3 ), 𝑓 (-𝑅 -𝑡 1 , 𝑅 -𝑡 2 , 𝑧 -𝑡 3 ), etc., when 𝑅 goes large. To this effect, it is convenient to introduce companion functions to 𝑓 , 𝑔, 𝑘 and ℓ as follows.

Definition 2. Let ⃗ 𝑡 = (𝑡 1 , 𝑡 2 , 𝑡 3 ) ∈ 𝒜. We define 𝐹 ⃗ 𝑡 , 𝐺 ⃗ 𝑡 , 𝐾 ⃗ 𝑡 and 𝐿 ⃗ 𝑡 from R to R by 𝐹 ⃗ 𝑡 (𝑅) = 𝑓 (𝑅 -𝑡 1 , 𝑅 -𝑡 2 , 𝑧 -𝑡 3 ), 𝐺 ⃗ 𝑡 (𝑅) = 𝑔(𝑅 -𝑡 1 , 𝑅 -𝑡 2 , 𝑧 -𝑡 3 ), 𝐾 ⃗ 𝑡 (𝑅) = 𝑘(𝑅 -𝑡 1 , 𝑅 -𝑡 2 , 𝑧 -𝑡 3 ), and 𝐿 ⃗ 𝑡 (𝑅) = ℓ(𝑅 -𝑡 1 , 𝑅 -𝑡 2 , 𝑧 -𝑡 3 ).
One easily checks the following proposition.

Proposition 2. For any

(𝑡 1 , 𝑡 2 ) ∈ [-𝑠, 𝑠] 2 , any 𝑡 3 > 0 and any 𝑅 ∈ R, 𝑓 (-𝑅 -𝑡 1 , 𝑅 -𝑡 2 , 𝑧 -𝑡 3 ) = 𝐹 (-𝑡 1 ,𝑡 2 ,𝑡 3 ) (𝑅), 𝑔(-𝑅 -𝑡 1 , 𝑅 -𝑡 2 , 𝑧 -𝑡 3 ) = 𝐺 (-𝑡 1 ,𝑡 2 ,𝑡 3 ) (𝑅), 𝑓 (-𝑅 -𝑡 1 , -𝑅 -𝑡 2 , 𝑧 -𝑡 3 ) = -𝐹 (-𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅), 𝑔(-𝑅 -𝑡 1 , -𝑅 -𝑡 2 , 𝑧 -𝑡 3 ) = 𝐺 (-𝑡 1 ,-𝑡 2 ,𝑡 3 (𝑅), 𝑓 (𝑅 -𝑡 1 , -𝑅 -𝑡 2 , 𝑧 -𝑡 3 ) = -𝐹 (𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅); 𝑔(𝑅 -𝑡 1 , -𝑅 -𝑡 2 , 𝑧 -𝑡 3 ) = 𝐺 (𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅); 𝑘(-𝑅 -𝑡 1 , 𝑅 -𝑡 2 , 𝑧 -𝑡 3 ) = -𝐾 (-𝑡 1 ,𝑡 2 ,𝑡 3 ) (𝑅), ℓ(-𝑅 -𝑡 1 , 𝑅 -𝑡 2 , 𝑧 -𝑡 3 ) = 𝐿 (-𝑡 1 ,𝑡 2 ,𝑡 3 ) (𝑅), 𝑘(-𝑅 -𝑡 1 , -𝑅 -𝑡 2 , 𝑧 -𝑡 3 ) = 𝐾 (-𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅), ℓ(-𝑅 -𝑡 1 , -𝑅 -𝑡 2 , 𝑧 -𝑡 3 ) = -𝐿 (-𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅), 𝑘(𝑅 -𝑡 1 , -𝑅 -𝑡 2 , 𝑧 -𝑡 3 ) = -𝐾 (𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅); ℓ(𝑅 -𝑡 1 , -𝑅 -𝑡 2 , 𝑧 -𝑡 3 ) = -𝐿 (𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅).
The essential ingredient for the proof of Equations ( 4) and ( 6) is to get asymptotic expansions of the functions 𝐹 ⃗ 𝑡 , 𝐺 ⃗ 𝑡 , 𝐾 ⃗ 𝑡 , 𝐿 ⃗ 𝑡 (with respect to powers of 1/𝑅), with explicit error bounds. The important point is that these error bounds are uniform with respect to the variable ⃗ 𝑡 ∈ 𝒜, allowing us to integrate them on 𝒜. Such expansions are given in Lemma 1 below. Before stating it, we need to introduce more notations and to recall some properties of homogeneous polynomials.

We fix once and for all two positive constants 𝜔 𝑠 and 𝜔 𝑧 such that 𝜔 𝑠 < 1, and we pose

𝐶 = max { 𝑠 𝜔 𝑠 , 𝑧 𝜔 𝑧 } , ( 15
)
hence 𝑠 𝐶 ≤ 𝜔 𝑠 < 1 and 𝑧 𝐶 ≤ 𝜔 𝑧 . From now on, we assume that 𝑅 ≥ 𝐶 with 𝐶 given by Equation (15).

Remark 1.

Introducing the rescaling quantities 𝜔 𝑠 and 𝜔 𝑧 is a means to assume that 𝑅 ≥ 𝐶 whatever the dimensions 𝑠 and 𝑟 of 𝒜 and the height 𝑧 of the measurement area (on can simply take 𝜔 𝑧 = 𝑧/𝑅 and 𝜔 𝑠 = 𝑠/𝑅, if 𝑅 > 𝑠). The assumption 𝑅 ≥ 𝐶 reflects of course the asymptotic character of the present study. Moreover, the quantities 𝜔 𝑠 and 𝜔 𝑧 conveniently allow us to specify how large 𝑅 should be relative to 𝑠 and 𝑧 for the error estimates in the expansions below to hold true (see Lemmas to come and Remark 2 at the end of the Appendix).

Let ⃗ 𝛾 = (𝛾 1 , 𝛾 2 , 𝛾 3 )∈ R 3 . If 𝑎 𝑛 (⃗ 𝛾) is a homogeneous polynomial of degree 𝑛 with real coefficients, i.e., 𝑎 𝑛 (⃗ 𝛾) = 𝑛 ∑ 𝑖=0 𝑛-𝑖 ∑ 𝑗=0 𝛼 𝑖,𝑗 𝛾 𝑖 1 𝛾 𝑗 2 𝛾 𝑛-𝑖-𝑗 3 ,
where 𝛼 𝑖,𝑗 ∈ R, we define the associated homogeneous polynomial 𝐴 𝑛 (𝑠, 𝑧) in the variables 𝑠 and 𝑧 by:

𝐴 𝑛 (𝑠, 𝑧) = 𝑛 ∑ 𝑖=0 𝑛-𝑖 ∑ 𝑗=0 |𝛼 𝑖,𝑗 | 𝑠 𝑖+𝑗 𝑧 𝑛-(𝑖+𝑗) . ( 16
)
Observe that whenever ⃗ 𝛾 ∈ [-𝑠, 𝑠] 2 × (0, 𝑧], we have |𝑎 𝑛 (⃗ 𝛾)| ≤ 𝐴 𝑛 (𝑠, 𝑧). In particular (recall from Section 2 that 𝒜 = [-𝑠, 𝑠] 2 × [0, 𝑟] and 𝑧 > 𝑟),

∀ ⃗ 𝑡 ∈ 𝒜, |𝑎 𝑛 (𝑡 1 , 𝑡 2 , 𝑧 -𝑡 3 )| ≤ 𝐴 𝑛 (𝑠, 𝑧). (17) Moreover, if Δ 𝑛 (𝑠, 𝑧) = 𝑛 ∑ 𝑘=0 𝑏 𝑘 𝑠 𝑘 𝑧 𝑛-𝑘
is a homogeneous polynomial of degree 𝑛 in 𝑠 and 𝑧 with 𝑏 𝑘 ∈ R, we define another associated homogeneous polynomial Δ 𝑛,1 (𝑠, 𝑧) of degree 𝑛 -1 by

Δ 𝑛,1 (𝑠, 𝑧) = (|𝑏 0 |𝜔 𝑧 )𝑧 𝑛-1 + 𝑛 ∑ 𝑘=1 (|𝑏 𝑘 |𝜔 𝑠 ) 𝑠 𝑘-1 𝑧 𝑛-𝑘 . ( 18
)
Notice that for any 𝜉 ∈ [0, 1 We can now state our first lemma. We postpone its proof to the appendix at the end of this document.

Lemma 1. For any ⃗ 𝑡 ∈ 𝒜 and any 𝑅 ≥ 𝐶, where 𝐶 is defined by Equation (15), it holds

𝐹 ⃗ 𝑡 (𝑅) = 1 √ 2 • 1 𝑅 2 + 5𝑡 1 -𝑡 2 2 √ 2 • 1 𝑅 3 + 33𝑡 2 1 -3𝑡 2 2 -6𝑡 1 𝑡 2 -10𝑡 2 3 + 20𝑧𝑡 3 -10𝑧 2 8 √ 2 • 1 𝑅 4 + 𝛿 1 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 5 𝐺 ⃗ 𝑡 (𝑅) = -1 √ 2 • 1 𝑅 - 𝑡 1 + 𝑡 2 2 √ 2 • 1 𝑅 2 - 𝑡 2 1 + 𝑡 2 2 + 6𝑡 1 𝑡 2 -2𝑡 2 3 + 4𝑧𝑡 3 -2𝑧 2 8 √ 2 • 1 𝑅 3 + 𝛿 2 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 4 𝐾 ⃗ 𝑡 (𝑅) = 𝜋 2(𝑧 -𝑡 3 ) - √ 2 1 𝑅 - √ 2(𝑡 1 + 𝑡 2 ) 2 • 1 𝑅 2 + 𝛿 3 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 , 𝐿 ⃗ 𝑡 (𝑅) = -arcsinh(1) + 𝑡 2 -𝑡 1 √ 2 • 1 𝑅 - 3𝑡 2 1 -𝑡 2 2 -2𝑡 1 𝑡 2 -2𝑡 2 3 + 4𝑧𝑡 3 -2𝑧 2 4 √ 2 • 1 𝑅 2 + 𝛿 4 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 ,
where Proof. From Lemma 1 we get

|𝛿 1 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ Δ (1) 3 (𝑠, 𝑧), |𝛿 2 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ Δ (2) 3 (𝑠, 𝑧), |𝛿 3 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ Δ (3
𝐹 ⃗ 𝑡 (𝑅) = 1 √ 2 • 1 𝑅 2 + 5𝑡 1 -𝑡 2 2 √ 2 • 1 𝑅 3 + 𝛿 5 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 4 , 𝐺 ⃗ 𝑡 (𝑅) = -1 √ 2 • 1 𝑅 - 𝑡 1 + 𝑡 2 2 √ 2 • 1 𝑅 2 + 𝛿 6 ( ⃗ 𝑡
|𝛿 5 ( ⃗ 𝑡, 𝑧, 𝑅)| = ⃒ ⃒ ⃒ ⃒ ⃒ 33𝑡 2 1 -3𝑡 2 2 -6𝑡 1 𝑡 2 -10(𝑧 -𝑡 3 ) 2 8 √ 2 + 𝛿 1 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 ⃒ ⃒ ⃒ ⃒ ⃒ ≤ 42𝑠 2 + 10𝑧 2 8 √ 2 + Δ (1) 3 (𝑠, 𝑧) 𝑅 . Now, since 1/𝑅 ∈ [0, 1/𝐶], Equation (19) implies that |Δ (1) 3 (𝑠, 𝑧)•1/𝑅| ≤ Δ (1) 3,1 (𝑠, 𝑧). Consequently, |𝛿 5 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ Δ (5) 2 (𝑠, 𝑧) where Δ (5) 2 (𝑠, 𝑧) = 42𝑠 2 +10𝑧 2 8 √ 2 + Δ (1)
3,1 (𝑠, 𝑧) is a homogeneous polynomial of degree 2. The result for 𝐺 ⃗ 𝑡 (𝑅) is obtained similarly.

Proof of Equation (4)

From Equation (3), we get on integrating the term 𝑥 1 𝜕 𝑥 1 by parts that :

- 4𝜋 𝜇 0 ∫∫ 𝑄 𝑅 𝑥 1 𝐵 3 [ ⃗ 𝑚, 𝑥 3 ](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = ∫ 𝑅 -𝑅 [ 𝑥 1 (𝑃 3 ⋆ ̃ 𝑚 1 -𝑃 1 ⋆ ̃ 𝑚 3 ) (⃗ 𝑥) ] 𝑅 𝑥 1 =-𝑅 d𝑥 2 + ∫ 𝑅 -𝑅 [ 𝑥 1 (𝑃 3 ⋆ ̃ 𝑚 2 -𝑃 2 ⋆ ̃ 𝑚 3 ) (⃗ 𝑥) ] 𝑅 𝑥 2 =-𝑅 d𝑥 1 - ∫∫ 𝑄 𝑅 𝑃 3 ⋆ ̃ 𝑚 1 (⃗ 𝑥) d𝑥 1 d𝑥 2 + ∫∫ 𝑄 𝑅 𝑃 1 ⋆ ̃ 𝑚 3 (⃗ 𝑥) d𝑥 1 d𝑥 2 .
Now, replacing 𝑃 1 , 𝑃 2 and 𝑃 3 by their expressions and using Fubini's theorem to interchange the integration on 𝑄 𝑅 and the integration on 𝒜 arising from the convolution, we get in view of Proposition 1 that

∫∫ 𝑄 𝑅 𝑥 1 𝐵 3 [ ⃗ 𝑚, 𝑥 3 ](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = - 𝜇 0 4 𝜋 ∫∫∫ 𝒜 ( 𝐼 1 ( ⃗ 𝑡) + 𝐼 2 ( ⃗ 𝑡) + 𝐼 3 ( ⃗ 𝑡) + 𝐼 4 ( ⃗ 𝑡) ) d ⃗ 𝑡, ( 20
)
where, for ⃗ 𝑡 ∈ 𝒜:

𝐼 1 ( ⃗ 𝑡) = [[ (𝑥 1 (𝑥 3 -𝑡 3 ) 𝑚 1 ( ⃗ 𝑡) -𝑥 1 (𝑥 1 -𝑡 1 ) 𝑚 3 ( ⃗ 𝑡)) 𝑓 (⃗ 𝑥 -⃗ 𝑡) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 , 𝐼 2 ( ⃗ 𝑡) = [ ((𝑥 3 -𝑡 3 ) 𝑚 2 ( ⃗ 𝑡) -(𝑥 2 -𝑡 2 ) 𝑚 3 ( ⃗ 𝑡)) ∫ 𝑅 -𝑅 𝑥 1 ‖⃗ 𝑥 -⃗ 𝑡‖ 3 d𝑥 1 ] 𝑅 𝑥 2 =-𝑅 , 𝐼 3 ( ⃗ 𝑡) = -(𝑥 3 -𝑡 3 ) 𝑚 1 ( ⃗ 𝑡) [[ 𝑘(⃗ 𝑥 -⃗ 𝑡) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 , 𝐼 4 ( ⃗ 𝑡) = 𝑚 3 ( ⃗ 𝑡) [[ ℓ(⃗ 𝑥 -⃗ 𝑡) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 .
To simplify 𝐼 2 ( ⃗ 𝑡) further, we can rewrite 𝑥 1 as (𝑥 1 -𝑡 1 ) + 𝑡 1 and ‖⃗ 𝑥 -⃗ 𝑡‖ as ‖(𝑥 2 -𝑡 2 , 𝑥 1 -𝑡 1 , 𝑥 3 -𝑡 3 )‖, which leads us to

∫ 𝑅 -𝑅 𝑥 1 ‖⃗ 𝑥 -⃗ 𝑡‖ 3 d𝑥 1 = [ 𝑔(⃗ 𝑥 -⃗ 𝑡) + 𝑡 1 𝑓 (𝑥 2 -𝑡 2 , 𝑥 1 -𝑡 1 , 𝑥 3 -𝑡 3 ) ] 𝑅 𝑥 1 =-𝑅 .
Now, grouping terms according to the parity of the powers of 𝑥 1 and 𝑥 2 , we obtain the following expressions for 𝐼 1 ( ⃗ 𝑡) and 𝐼 2 ( ⃗ 𝑡):

𝐼 1 ( ⃗ 𝑡) = ((𝑥 3 -𝑡 3 ) 𝑚 1 ( ⃗ 𝑡) + 𝑡 1 𝑚 3 ( ⃗ 𝑡)) [[ 𝑥 1 𝑓 (⃗ 𝑥 -⃗ 𝑡) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 -𝑚 3 ( ⃗ 𝑡) [[ 𝑥 2 1 𝑓 (⃗ 𝑥 -⃗ 𝑡) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 , 𝐼 2 ( ⃗ 𝑡) = ((𝑥 3 -𝑡 3 ) 𝑚 2 ( ⃗ 𝑡) + 𝑡 2 𝑚 3 ( ⃗ 𝑡)) [[ 𝑔(⃗ 𝑥 -⃗ 𝑡) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 + 𝑡 1 ((𝑥 3 -𝑡 3 ) 𝑚 2 ( ⃗ 𝑡) + 𝑡 2 𝑚 3 ( ⃗ 𝑡)) [[ 𝑓 (𝑥 2 -𝑡 2 , 𝑥 1 -𝑡 1 , 𝑥 3 -𝑡 3 ) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 -𝑚 3 ( ⃗ 𝑡) [[ 𝑥 2 𝑔(⃗ 𝑥 -⃗ 𝑡) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 -𝑡 1 𝑚 3 [[ 𝑥 2 𝑓 (𝑥 2 -𝑡 2 , 𝑥 1 -𝑡 1 , 𝑥 3 -𝑡 3 ) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅
.

From now on, we assume that 𝑥 3 is fixed and equal to 𝑧 and that the hypotheses of Lemma 1 are satisfied.

Asymptotic expansion of 𝐼 1 . Using Proposition 2 and Lemma 1, we get

[[ 𝑥 1 𝑓 (⃗ 𝑥 -⃗ 𝑡) |𝑥 3 =𝑧 ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 = 𝑅 ( 𝐹 ⃗ 𝑡 (𝑅) + 𝐹 (-𝑡 1 ,𝑡 2 ,𝑡 3 ) (𝑅) + 𝐹 (𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅) + 𝐹 (-𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅) ) = 4 √ 2 • 1 𝑅 + 𝛿 7 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 , [[ 𝑥 2 1 𝑓 (⃗ 𝑥 -⃗ 𝑡) |𝑥 3 =𝑧 ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 = 𝑅 2 [[ 𝑓 (⃗ 𝑥 -⃗ 𝑡) |𝑥 3 =𝑧 ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 = 𝑅 2 ( 𝐹 ⃗ 𝑡 (𝑅) -𝐹 (-𝑡 1 ,𝑡 2 ,𝑡 3 ) (𝑅) + 𝐹 (𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅) -𝐹 (-𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅) ) = 10𝑡 1 √ 2 • 1 𝑅 + 𝛿 8 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 ,
where |𝛿 7 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ 4Δ

3 (𝑠, 𝑧). Therefore,

𝐼 1 ( ⃗ 𝑡) = 4(𝑧 -𝑡 3 ) 𝑚 1 ( ⃗ 𝑡) -6𝑡 1 𝑚 3 ( ⃗ 𝑡) 𝑅 √ 2 + 𝛿 9 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 (21)
where

|𝛿 9 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ 4𝑧|𝑚 1 ( ⃗ 𝑡)|Δ (5) 2 (𝑠, 𝑧) + 4|𝑚 3 ( ⃗ 𝑡)|(𝑠Δ (5) 2 (𝑠, 𝑧) + Δ (1)
3 (𝑠, 𝑧)).

Asymptotic expansion of 𝐼 2 .

A simple interchange between variables 𝑥 1 and 𝑥 2 and between variables 𝑡 1 and 𝑡 2 in the computations of the previous paragraph shows that

[[ 𝑓 (𝑥 2 -𝑡 2 , 𝑥 1 -𝑡 1 , 𝑧 -𝑡 3 ) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 = 10𝑡 2 √ 2 • 1 𝑅 3 + 𝛿 8 ((𝑡 2 , 𝑡 1 , 𝑡 3 ), 𝑧, 𝑅) 𝑅 5 , = 𝛿 10 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 , [[ 𝑥 2 𝑓 (𝑥 2 -𝑡 2 , 𝑥 1 -𝑡 1 , 𝑧 -𝑡 3 ) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 = 4 √ 2 • 1 𝑅 + 𝛿 7 ((𝑡 2 , 𝑡 1 , 𝑡 3 ), 𝑧, 𝑅) 𝑅 3 ,
where

|𝛿 10 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ 10𝑠 √ 2 + 4Δ (1)
3 (𝑠, 𝑧)

1 (𝑠, 𝑧) = 10𝑠 √ 2 + 4Δ (1)
3,2 (𝑠, 𝑧). Here, Δ

3,2 is the polynomial constructed from Δ

(1)

3 by two successive applications of the process defined by Equation (18).

Moreover, using Proposition 2 and Lemma 1 on 𝑔 ⃗ 𝑡 , we get

[[ 𝑔(⃗ 𝑥 -⃗ 𝑡) |𝑥 3 =𝑧 ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 = 𝐺 ⃗ 𝑡 (𝑅) -𝐺 (-𝑡 1 ,𝑡 2 ,𝑡 3 ) (𝑅) -𝐺 (𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅) + 𝐺 (-𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅) = 𝛿 11 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 , [[ 𝑥 2 𝑔(⃗ 𝑥 -⃗ 𝑡) |𝑥 3 =𝑧 ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 = 𝑅 ( 𝐺 ⃗ 𝑡 (𝑅) -𝐺 (-𝑡 1 ,𝑡 2 ,𝑡 3 ) (𝑅) + 𝐺 (𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅) -𝐺 (-𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅) ) = -2𝑡 1 √ 2 • 1 𝑅 + 𝛿 12 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 ,
where

|𝛿 11 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ 4Δ (6)
2 (𝑠, 𝑧) and

|𝛿 12 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ 4Δ (2)
3 (𝑠, 𝑧). Therefore,

𝐼 2 ( ⃗ 𝑡) = - 2𝑡 1 𝑚 3 ( ⃗ 𝑡) 𝑅 √ 2 + 𝛿 13 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 (22)
where

|𝛿 13 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ (4Δ (6)
2 (𝑠, 𝑧) + 𝑠Δ

(10) 1 (𝑠, 𝑧)) 𝑧|𝑚 2 ( ⃗ 𝑡)| + (4𝑠Δ (6) 2 (𝑠, 𝑧) + 4Δ (2) 3 (𝑠, 𝑧) + 4𝑠Δ (5) 2 (𝑠, 𝑧) + 𝑠 2 Δ (10) 1 (𝑠, 𝑧)) 𝑚 3 ( ⃗ 𝑡)
Asymptotic expansions of 𝐼 3 and 𝐼 4 . Following the same line of argument as we used for 𝐼 1 and 𝐼 2 , we get

[[ 𝑘(⃗ 𝑥 -⃗ 𝑡) |𝑥 3 =𝑧 ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 = 𝐾 ⃗ 𝑡 (𝑅) + 𝐾 (-𝑡 1 ,𝑡 2 ,𝑡 3 ) (𝑅) + 𝐾 (𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅) + 𝐾 (-𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅) = 2𝜋 𝑧 -𝑡 3 -4 √ 2 1 𝑅 + 𝛿 14 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 , [[ ℓ(⃗ 𝑥 -⃗ 𝑡) |𝑥 3 =𝑧 ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 = 𝐿 ⃗ 𝑡 (𝑅) -𝐿 (-𝑡 1 ,𝑡 2 ,𝑡 3 ) (𝑅) + 𝐿 (𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅) -𝐿 (-𝑡 1 ,-𝑡 2 ,𝑡 3 ) (𝑅) = -4𝑡 1 √ 2 • 1 𝑅 + 𝛿 15 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 ,
where |𝛿 14 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ 4Δ

3 (𝑠, 𝑧). Therefore,

𝐼 3 ( ⃗ 𝑡) = -2𝜋𝑚 1 ( ⃗ 𝑡) + 4(𝑧 -𝑡 3 ) √ 2 𝑚 1 ( ⃗ 𝑡) 𝑅 + 𝛿 16 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 , ( 23
) 𝐼 4 ( ⃗ 𝑡) = -4𝑡 1 𝑚 3 ( ⃗ 𝑡) 𝑅 √ 2 + 𝛿 17 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 , ( 24
)
where |𝛿 16 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ 4Δ

∫∫ 𝑄 𝑅 𝑥 1 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 𝜇 0 2 ⟨𝑚 1 ⟩ + 3𝜇 0 𝜋𝑅 √ 2 (⟨𝑡 1 𝑚 3 ⟩ + ⟨𝑡 3 𝑚 1 ⟩ -𝑧⟨𝑚 1 ⟩) + 𝛿 18 (𝑠,
) 𝑧⟨|𝑚 1 |⟩ + ( 4Δ (6)
2 (𝑠, 𝑧) + 𝑠Δ [START_REF] Weiss | Paleomagnetic analysis using SQUID microscopy[END_REF] 1 (𝑠, 𝑧)

) 𝑧⟨|𝑚 2 |⟩ + ( 4(Δ (1)
3 (𝑠, 𝑧) + Δ

(2)

3 (𝑠, 𝑧) + Δ (4) 3 (𝑠, 𝑧)) + 4𝑠(2Δ (5) 2 (𝑠, 𝑧) + Δ (6) 2 (𝑠, 𝑧)) + 𝑠 2 Δ (10) 1 (𝑠, 𝑧) ) ⟨|𝑚 3 |⟩ )
This shows that 𝛿 18 is bounded by a quantity that depends only on 𝑠 and 𝑧 but not on 𝑅.

Proof of Equation (6)

We follow the path that led us to Equation (4): starting from Equation (3) we get

- 2 𝜇 0 ∫∫ 𝑄 𝑅 𝑅 𝐵 3 [ ⃗ 𝑚, 𝑥 3 ](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = ∫ 𝑅 -𝑅 𝑅 [ (𝑃 3 ⋆ ̃ 𝑚 1 -𝑃 1 ⋆ ̃ 𝑚 3 ) (⃗ 𝑥) ] 𝑅 𝑥 1 =-𝑅 d𝑥 2 + ∫ 𝑅 -𝑅 𝑅 [ (𝑃 3 ⋆ ̃ 𝑚 2 -𝑃 2 ⋆ ̃ 𝑚 3 ) (⃗ 𝑥) ] 𝑅 𝑥 2 =-𝑅 d𝑥 1 .
Now, replacing 𝑃 1 , 𝑃 2 and 𝑃 3 by their expressions and using Fubini, we obtain

∫∫ 𝑄 𝑅 𝑅 𝐵 3 [ ⃗ 𝑚, 𝑥 3 ](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = - 𝜇 0 4 𝜋 ∫∫∫ 𝒜 ( 𝐼 5 ( ⃗ 𝑡) + 𝐼 6 ( ⃗ 𝑡) ) d ⃗ 𝑡, (25)
where:

𝐼 5 ( ⃗ 𝑡) = 𝑅 [[ ((𝑥 3 -𝑡 3 ) 𝑚 1 ( ⃗ 𝑡) -(𝑥 1 -𝑡 1 ) 𝑚 3 ( ⃗ 𝑡)) 𝑓 (⃗ 𝑥 -⃗ 𝑡) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 = 𝑅((𝑥 3 -𝑡 3 ) 𝑚 1 ( ⃗ 𝑡) + 𝑡 1 𝑚 3 ( ⃗ 𝑡)) [[ 𝑓 (⃗ 𝑥 -⃗ 𝑡) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 -𝑅 𝑚 3 ( ⃗ 𝑡) [[ 𝑥 1 𝑓 (⃗ 𝑥 -⃗ 𝑡) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 , 𝐼 6 ( ⃗ 𝑡) = 𝑅 [ ((𝑥 3 -𝑡 3 ) 𝑚 2 ( ⃗ 𝑡) -(𝑥 2 -𝑡 2 ) 𝑚 3 ( ⃗ 𝑡)) ∫ 𝑅 -𝑅 1 ‖⃗ 𝑥 -⃗ 𝑡‖ d𝑥 1 ] 𝑅 𝑥 2 =-𝑅 = 𝑅 [[ ((𝑥 3 -𝑡 3 ) 𝑚 2 ( ⃗ 𝑡) -(𝑥 2 -𝑡 2 ) 𝑚 3 ( ⃗ 𝑡)) 𝑓 (𝑥 2 -𝑡 2 , 𝑥 1 -𝑡 1 , 𝑥 3 -𝑡 3 ) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 = 𝑅((𝑥 3 -𝑡 3 ) 𝑚 2 ( ⃗ 𝑡) + 𝑡 2 𝑚 3 ( ⃗ 𝑡)) [[ 𝑓 (𝑥 2 -𝑡 2 , 𝑥 1 -𝑡 1 , 𝑥 3 -𝑡 3 ) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 -𝑅 𝑚 3 ( ⃗ 𝑡) [[ 𝑥 2 𝑓 (𝑥 2 -𝑡 2 , 𝑥 1 -𝑡 1 , 𝑥 3 -𝑡 3 ) ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅
. Now, using the same arguments as in Section 4.2, and assuming as before that 𝑥 3 is fixed and equal to 𝑧 and that the hypotheses of Lemma 1 are satisfied, we see that

𝑅 [[ 𝑓 (⃗ 𝑥 -⃗ 𝑡) |𝑥 3 =𝑧 ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 = 10𝑡 1 √ 2 • 1 𝑅 2 + 𝛿 19 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 , 𝑅 [[ 𝑥 1 𝑓 (⃗ 𝑥 -⃗ 𝑡) |𝑥 3 =𝑧 ] 𝑅 𝑥 1 =-𝑅 ] 𝑅 𝑥 2 =-𝑅 = 4 √ 2 + 33𝑡 2 1 -3𝑡 2 2 -10(𝑧 -𝑡 3 ) 2 2 √ 2 • 1 𝑅 2 +
𝛿 20 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 ,

where |𝛿 19 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ 4Δ (1)
3 (𝑠, 𝑧) 1 𝑅 and 𝛿 20 ( ⃗ 𝑡, 𝑧, 𝑅) ≤ 4Δ

(1)

3 (𝑠, 𝑧). Therefore,

𝐼 5 ( ⃗ 𝑡) = -4𝑚 3 ( ⃗ 𝑡) √ 2 + 20(𝑧 -𝑡 3 )𝑡 1 𝑚 1 ( ⃗ 𝑡) -13𝑡 2 1 𝑚 3 ( ⃗ 𝑡) + 3𝑡 2 2 𝑚 3 ( ⃗ 𝑡) + 10(𝑧 -𝑡 3 ) 2 𝑚 3 ( ⃗ 𝑡) 2𝑅 2 √ 2 + 𝛿 21 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 ,
where |𝛿 21 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ 4Δ

(1)

3 (𝑠, 𝑧)

( |𝑚 3 ( ⃗ 𝑡)|(1 + 𝑠 𝑅 ) + |𝑚 1 ( ⃗ 𝑡)| 𝑧 𝑅 )
and therefore, when 𝑅 ≥ 𝐶,

|𝛿 21 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ 4Δ (1)
3 (𝑠, 𝑧)

( |𝑚 3 ( ⃗ 𝑡)|(1 + 𝜔 𝑠 ) + |𝑚 1 ( ⃗ 𝑡)|𝜔 𝑧 )
by Equation (15). Accordingly,

𝐼 6 ( ⃗ 𝑡) = -4𝑚 3 ( ⃗ 𝑡) √ 2 + 20(𝑧 -𝑡 3 )𝑡 2 𝑚 2 ( ⃗ 𝑡) -13𝑡 2 2 𝑚 3 ( ⃗ 𝑡) + 3𝑡 2 1 𝑚 3 ( ⃗ 𝑡) + 10(𝑧 -𝑡 3 ) 2 𝑚 3 ( ⃗ 𝑡) 2𝑅 2 √ 2 + 𝛿 22 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3 ,
where

|𝛿 22 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ 4Δ (1)
3 (𝑠, 𝑧)

( |𝑚 3 ( ⃗ 𝑡)|(1 + 𝜔 𝑠 ) + |𝑚 2 ( ⃗ 𝑡)|𝜔 𝑧 )
. Putting these last two equations together we finally get

∫∫ 𝑄 𝑅 𝑅 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 2𝜇 0 𝜋 √ 2 ⟨𝑚 3 ⟩ + 5𝜇 0 4𝜋𝑅 2 √ 2 ( -2𝑧 2 ⟨𝑚 3 ⟩ + 𝑧 (4⟨𝑡 3 𝑚 3 ⟩ -2⟨𝑡 1 𝑚 1 ⟩ -2⟨𝑡 2 𝑚 2 ⟩) + ⟨𝑡 2 1 𝑚 3 ⟩ + ⟨𝑡 2 2 𝑚 3 ⟩ -2⟨𝑡 2 3 𝑚 3 ⟩ + 2⟨𝑡 1 𝑡 3 𝑚 1 ⟩ + 2⟨𝑡 2 𝑡 3 𝑚 2 ⟩ ) + 𝛿 23 (𝑠, 𝑧, 𝑅) 𝑅 3 ,
where

𝛿 23 (𝑠, 𝑧, 𝑅) = - 𝜇 0 4𝜋 ∫∫∫ 𝒜 ( 𝛿 21 ( ⃗ 𝑡, 𝑧, 𝑅) + 𝛿 22 ( ⃗ 𝑡, 𝑧, 𝑅) )
d ⃗ 𝑡. Now, 𝛿 23 is bounded when 𝑅 goes to infinity, since for any 𝑅 ≥ 𝐶 (where 𝐶 is the constant given by Equation ( 15)) we have

|𝛿 23 (𝑠, 𝑧, 𝑅)| ≤ 𝜇 0 𝜋 Δ (1)
3 (𝑠, 𝑧)

( 2⟨|𝑚 3 |⟩(1 + 𝜔 𝑠 ) + ⟨|𝑚 1 |⟩𝜔 𝑧 + ⟨|𝑚 2 |⟩𝜔 𝑧 )
.

5 Proofs of the remaining equations

Generalities

Consider a linear isometry Ψ of R 2 and define the linear isometry Ψ of R 3 by Ψ(𝑥 1 , 𝑥 2 , 𝑥 3 ) = (Ψ(𝑥 1 , 𝑥 2 ), 𝑥 3 ). We define 𝑄 ′ 𝑅 = Ψ(𝑄 𝑅 ). Denoting by Ψ 𝑖 the 𝑖-th component of Ψ (where 𝑖 ∈ {1, 2}), we have, by the change of variable

(𝑥 ′ 1 , 𝑥 ′ 2 ) = Ψ(𝑥 1 , 𝑥 2 ), ∫∫ 𝑄 ′ 𝑅 𝑥 ′ 𝑖 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 ′ 1 , 𝑥 ′ 2 ) d𝑥 ′ 1 d𝑥 ′ 2 = ∫∫ 𝑄 𝑅 Ψ 𝑖 (⃗ 𝑥) 𝐵 3 [ ⃗ 𝑚, 𝑧](Ψ(𝑥 1 , 𝑥 2 )) d𝑥 1 d𝑥 2 . ( 26
)
Moreover, according to Equation (1), we have, for any

⃗ 𝑥 ∈ R 3 such that 𝑥 3 > 𝑟, 𝐵 3 [ ⃗ 𝑚, 𝑥 3 ](Ψ(𝑥 1 , 𝑥 2 )) = -𝜇 0 4𝜋 𝜕 𝑥 3 ( ∫∫∫ R 3 ⟨ ̃ ⃗ 𝑚( ⃗ 𝑡 ′ ), Ψ(⃗ 𝑥) -⃗ 𝑡 ′ ⟩ ‖ Ψ(⃗ 𝑥) -⃗ 𝑡 ′ ‖ 3 d ⃗ 𝑡 ′ )
.

Then, using the change of variable ⃗ 𝑡 ′ = Ψ( ⃗ 𝑡) and the fact that Ψ is a linear isometry (and hence preserves the inner product and the norm), the above expression becomes

-𝜇 0 4𝜋 𝜕 𝑥 3 ( ∫∫∫ R 3 ⟨ Ψ-1 ( ̃ ⃗ 𝑚( Ψ( ⃗ 𝑡)) ) , ⃗ 𝑥 -⃗ 𝑡⟩ ‖⃗ 𝑥 -⃗ 𝑡‖ 3 d ⃗ 𝑡 ) . Finally, defining 𝑀 1 , 𝑀 2 and 𝑀 3 by ⃗ 𝑀 ( ⃗ 𝑡) = Ψ-1 ( ̃ ⃗ 𝑚( Ψ( ⃗ 𝑡))
)

, we observe that

𝐵 3 [ ⃗ 𝑚, 𝑥 3 ](Ψ(𝑥 1 , 𝑥 2 )) = 𝐵 3 [ ⃗ 𝑀 , 𝑥 3 ](𝑥 1 , 𝑥 2 ). ( 27
)
Now, putting together Equations ( 26) and ( 27) and using the linearity of Ψ 1 and Ψ 2 , we conclude that

⎛ ⎜ ⎜ ⎜ ⎝ ∫∫ 𝑄 ′ 𝑅 𝑥 ′ 1 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 ′ 1 , 𝑥 ′ 2 ) d𝑥 ′ 1 d𝑥 ′ 2 ∫∫ 𝑄 ′ 𝑅 𝑥 ′ 2 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 ′ 1 , 𝑥 ′ 2 ) d𝑥 ′ 1 d𝑥 ′ 2 ⎞ ⎟ ⎟ ⎟ ⎠ = Ψ ⎛ ⎜ ⎜ ⎜ ⎝ ∫∫ 𝑄 𝑅 𝑥 1 𝐵 3 [ ⃗ 𝑀 , 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 ∫∫ 𝑄 𝑅 𝑥 2 𝐵 3 [ ⃗ 𝑀 , 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 ⎞ ⎟ ⎟ ⎟ ⎠ . ( 28
)
Accordingly,

∫∫ 𝑄 ′ 𝑅 𝑅 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 ′ 1 , 𝑥 ′ 2 ) d𝑥 ′ 1 d𝑥 ′ 2 = ∫∫ 𝑄 𝑅 𝑅 𝐵 3 [ ⃗ 𝑀 , 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 . ( 29
)
We now express certain moments of 𝑀 1 , 𝑀 2 and 𝑀 3 . Let 𝑖 ∈ {1, 2, 3} and let us denote by Ψ-1 𝑖 the 𝑖-th component of Ψ-1 . Also, we consider an arbitrary bounded function 𝜃 : R → R. Using successively the definition of 𝑀 𝑖 , the change of variable ⃗ 𝑡 ′ = Ψ( ⃗ 𝑡) together with the fact that Ψ is a linear isometry, the fact that Ψ-1 3 ( ⃗ 𝑡 ′ ) = 𝑡 ′ 3 and the linearity of Ψ-1 𝑖 , we get

⟨𝜃(𝑡 3 ) 𝑀 𝑖 ⟩ = ∫∫∫ R 3 𝜃(𝑡 3 ) Ψ-1 𝑖 ( ̃ ⃗ 𝑚( Ψ( ⃗ 𝑡))) d ⃗ 𝑡 = ∫∫∫ R 3 𝜃( Ψ-1 3 ( ⃗ 𝑡 ′ )) Ψ-1 𝑖 ( ̃ ⃗ 𝑚( ⃗ 𝑡 ′ )) d ⃗ 𝑡 ′ = Ψ-1 𝑖 ⎛ ⎜ ⎝ ⟨𝜃(𝑡 3 ) 𝑚 1 ⟩ ⟨𝜃(𝑡 3 ) 𝑚 2 ⟩ ⟨𝜃(𝑡 3 ) 𝑚 3 ⟩ ⎞ ⎟ ⎠ .
Observing that, for any ⃗ 𝑥 ∈ R 3 , Ψ-1 (⃗ 𝑥) = (Ψ -1 (𝑥 1 , 𝑥 2 ), 𝑥 3 ), we thus obtain:

( ⟨𝜃(𝑡 3 ) 𝑀 1 ⟩ ⟨𝜃(𝑡 3 ) 𝑀 2 ⟩ ) = Ψ -1 ( ⟨𝜃(𝑡 3 ) 𝑚 1 ⟩ ⟨𝜃(𝑡 3 ) 𝑚 2 ⟩ ) and ⟨𝜃(𝑡 3 ) 𝑀 3 ⟩ = ⟨𝜃(𝑡 3 ) 𝑚 3 ⟩. ( 30
)
Accordingly, we obtain ⟨𝑡 𝑖 𝜃(𝑡 3 )

𝑀 3 ⟩ = ∫∫∫ R 3 Ψ-1 𝑖 ( ⃗ 𝑡 ′ ) 𝜃(𝑡 ′ 3 ) ̃ 𝑚 3 ( ⃗ 𝑡 ′ ) d ⃗ 𝑡 ′ = Ψ-1 𝑖 ⎛ ⎜ ⎝ ⟨𝑡 1 𝜃(𝑡 3 ) 𝑚 3 ⟩ ⟨𝑡 2 𝜃(𝑡 3 ) 𝑚 3 ⟩ ⟨𝑡 3 𝜃(𝑡 3 ) 𝑚 3 ⟩ ⎞ ⎟ ⎠, whence ( ⟨𝑡 1 𝜃(𝑡 3 ) 𝑀 3 ⟩ ⟨𝑡 2 𝜃(𝑡 3 ) 𝑀 3 ⟩ ) = Ψ -1 ( ⟨𝑡 1 𝜃(𝑡 3 ) 𝑚 3 ⟩ ⟨𝑡 2 𝜃(𝑡 3 ) 𝑚 3 ⟩ ) and ⟨𝑡 3 𝜃(𝑡 3 ) 𝑀 3 ⟩ = ⟨𝑡 3 𝜃(𝑡 3 ) 𝑚 3 ⟩. (31)

Proof of Equation (5)

To prove Equation ( 5), we define Ψ : ⃗ 𝑥 ↦ → (𝑥 2 , 𝑥 1 ). It is a linear involutive isometry, hence Equation (28) gives in this context

∫∫ 𝑄 ′ 𝑅 𝑥 ′ 2 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 ′ 1 , 𝑥 ′ 2 ) d𝑥 ′ 1 d𝑥 ′ 2 = ∫∫ 𝑄 𝑅 𝑥 1 𝐵 3 [ ⃗ 𝑀 , 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 .
Now, observing that 𝑄 ′ 𝑅 = 𝑄 𝑅 and applying Equation (4) we get

∫∫ 𝑄 𝑅 𝑥 2 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 𝜇 0 2 ⟨𝑀 1 ⟩ + 3𝜇 0 𝜋𝑅 √ 2 (⟨𝑡 1 𝑀 3 ⟩ + ⟨𝑡 3 𝑀 1 ⟩ -𝑧⟨𝑀 1 ⟩) + 𝒪 ( 1 𝑅 3 )
.

We conclude by remarking that 30), (31) with 𝜃(𝑡 3 ) = 1 and Equation (30) with 𝜃(𝑡 3 ) = 𝑡 3 • 𝜒 [0,𝑧] (𝑡 3 ). [START_REF] Lima | Fast inversion of magnetic field maps of unidirectional planar geological magnetization[END_REF], ( 8) and [START_REF] Ponomarev | Some inverse problems with partial data[END_REF] We define Ψ as the rotation of angle 𝜋/4 and we apply Equation (28

⟨𝑀 1 ⟩ = ⟨𝑚 2 ⟩, ⟨𝑡 1 𝑀 3 ⟩ = ⟨𝑡 2 𝑚 3 ⟩ and ⟨𝑡 3 𝑀 1 ⟩ = ⟨𝑡 3 𝑚 2 ⟩ thanks to Equations (

Proofs of Equations

) to 𝑄 √ 2 2 𝑅 , so that 𝑄 ′ √ 2 2 𝑅 = 𝑆 𝑅 .
In this context, and according to Equations ( 4) and ( 5), Equation (28) becomes

⎛ ⎜ ⎜ ⎜ ⎝ ∫∫ 𝑆 𝑅 𝑥 1 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 ∫∫ 𝑆 𝑅 𝑥 2 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 ⎞ ⎟ ⎟ ⎟ ⎠ = Ψ ⎛ ⎜ ⎜ ⎜ ⎝ 𝜇 0 2 ⟨𝑀 1 ⟩ + 3𝜇 0 𝜋𝑅 (⟨𝑡 1 𝑀 3 ⟩ + ⟨𝑡 3 𝑀 1 ⟩ -𝑧⟨𝑀 1 ⟩) + 𝒪 ( 1 𝑅 3 ) 𝜇 0 2 ⟨𝑀 2 ⟩ + 3𝜇 0 𝜋𝑅 (⟨𝑡 2 𝑀 3 ⟩ + ⟨𝑡 3 𝑀 2 ⟩ -𝑧⟨𝑀 2 ⟩) + 𝒪 ( 1 𝑅 3 ) ⎞ ⎟ ⎟ ⎟ ⎠
.

Using the linearity of Ψ, the right hand side of the above expression becomes

𝜇 0 2 Ψ ( ⟨𝑀 1 ⟩ ⟨𝑀 2 ⟩ ) + 3𝜇 0 𝜋𝑅 ( Ψ ( ⟨𝑡 1 𝑀 3 ⟩ ⟨𝑡 2 𝑀 3 ⟩ ) + Ψ ( ⟨𝑡 3 𝑀 1 ⟩ ⟨𝑡 3 𝑀 2 ⟩ ) -𝑧 Ψ ( ⟨𝑀 1 ⟩ ⟨𝑀 2 ⟩ )) + 𝒪 ( 1 𝑅 3 )
, which directly gives Equations ( 7) and ( 8) thanks to Equations ( 30) and (31) with 𝜃(𝑡 3 ) = 1 and Equation (30) with 𝜃(𝑡 3 ) = 𝑡 3 • 𝜒 [0,𝑠] (𝑡 3 ). Now, together with Equation ( 6), Equation (29) gives us

∫∫ 𝑆 𝑅 𝑅 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 2𝜇 0 𝜋 ⟨𝑀 3 ⟩ + 5𝜇 0 2𝜋𝑅 2 (-2𝑧 2 ⟨𝑀 3 ⟩ + 𝑧 (4⟨𝑡 3 𝑀 3 ⟩ -2⟨𝑡 1 𝑀 1 ⟩ -2⟨𝑡 2 𝑀 2 ⟩) + ⟨𝑡 2 1 𝑀 3 ⟩ + ⟨𝑡 2 2 𝑀 3 ⟩ -2⟨𝑡 2 3 𝑀 3 ⟩ + 2⟨𝑡 1 𝑡 3 𝑀 1 ⟩ + 2⟨𝑡 2 𝑡 3 𝑀 2 ⟩) + 𝒪 ( 1 𝑅 3 ) .
Notice that (using again the change of variable ⃗ 𝑡 = Ψ-1 ( ⃗ 𝑡 ′ ) and the fact that Ψ is a linear isometry)

⟨𝑡 2 1 𝑀 3 ⟩ + ⟨𝑡 2 2 𝑀 3 ⟩ + ⟨𝑡 2 3 𝑀 3 ⟩ = ⟨(𝑡 2 1 + 𝑡 2 2 + 𝑡 2 3 ) 𝑀 3 ⟩ = ∫∫∫ R 3 ‖ ⃗ 𝑡‖ 2 ̃ 𝑚 3 ( Ψ( ⃗ 𝑡)) d ⃗ 𝑡 = ∫∫∫ R 3 ‖ Ψ-1 ( ⃗ 𝑡 ′ )‖ 2 ̃ 𝑚 3 ( ⃗ 𝑡 ′ ) d ⃗ 𝑡 ′ = ∫∫∫ R 3 ‖ ⃗ 𝑡 ′ ‖ 2 ̃ 𝑚 3 ( ⃗ 𝑡 ′ ) d ⃗ 𝑡 ′ = ⟨𝑡 2 1 𝑚 3 ⟩ + ⟨𝑡 2 2 𝑚 3 ⟩ + ⟨𝑡 2 3 𝑚 3 ⟩.
Combining this with Equation (30) where we set 𝜃(𝑡 3 ) = 𝑡 2 3 • 𝜒 [0,𝑠] (𝑡 3 ), we get

⟨𝑡 2 1 𝑀 3 ⟩ + ⟨𝑡 2 2 𝑀 3 ⟩ -2⟨𝑡 2 3 𝑀 3 ⟩ = ⟨𝑡 2 1 𝑚 3 ⟩ + ⟨𝑡 2 2 𝑚 3 ⟩ -2⟨𝑡 2 3 𝑚 3 ⟩.
Also, if we denote by 𝐴 the matrix of Ψ -1 in the canonical basis and by 𝐶 𝑀 (resp. 𝐶 𝑚 ) the 3 × 3 matrix whose element (𝑖, 𝑗) is ⟨𝑡 𝑖 𝜃(𝑡 3 ) 𝑀 𝑗 ⟩ (resp. ⟨𝑡 𝑖 𝜃(𝑡 3 ) 𝑚 𝑗 ⟩), we observe that

𝐶 𝑀 = 𝐴 𝐶 𝑚 𝐴 𝑇 . ( 32
)
Indeed, if 𝑖 and 𝑗 belong to {1, 2, 3}, then upon using the same change of variable we get that

⟨𝑡 𝑖 𝜃(𝑡 3 ) 𝑀 𝑗 ⟩ = ∫∫∫ R 3 𝑡 𝑖 𝜃(𝑡 3 ) Ψ-1 𝑗 ( ̃ ⃗ 𝑚( Ψ( ⃗ 𝑡))) d ⃗ 𝑡 = ∫∫∫ R 3 Ψ-1 𝑖 ( ⃗ 𝑡 ′ ) 𝜃(𝑡 ′ 3 ) Ψ-1 𝑗 ( ̃ ⃗ 𝑚( ⃗ 𝑡 ′ )) d ⃗ 𝑡 ′ ,
Denoting by 𝐴 𝑖 and 𝐴 𝑗 the 𝑖-th and 𝑗-th row of 𝐴 respectively, we see that

Ψ-1 𝑖 ( ⃗ 𝑡 ′ ) Ψ-1 𝑗 ( ̃ ⃗ 𝑚( ⃗ 𝑡 ′ )) = (𝐴 𝑖 ⃗ 𝑡 ′ )(𝐴 𝑗 ̃ ⃗ 𝑚( ⃗ 𝑡 ′ )) = (𝐴 𝑖 ⃗ 𝑡 ′ )(𝐴 𝑗 ̃ ⃗ 𝑚( ⃗ 𝑡 ′ )) 𝑇 = 𝐴 𝑖 ( ⃗ 𝑡 ′ ̃ ⃗ 𝑚( ⃗ 𝑡 ′ ) 𝑇 ) 𝐴 𝑇 𝑗 .
Therefore, ⟨𝑡 𝑖 𝜃(𝑡 3 ) 𝑀 𝑗 ⟩ = 𝐴 𝑖 𝐶 𝑚 𝐴 𝑇 𝑗 whence Equation (32) holds. Now, Ψ -1 being an isometry we have 𝐴 𝑇 = 𝐴 -1 , and therefore

⟨𝑡 1 𝜃(𝑡 3 ) 𝑀 1 ⟩ + ⟨𝑡 2 𝜃(𝑡 3 ) 𝑀 2 ⟩ + ⟨𝑡 3 𝜃(𝑡 3 ) 𝑀 3 ⟩ = tr (𝐴 𝐶 𝑚 𝐴 𝑇 ) = tr (𝐴 𝑇 𝐴 𝐶 𝑚 ) = tr (𝐶 𝑚 ) = ⟨𝑡 1 𝜃(𝑡 3 ) 𝑚 1 ⟩ + ⟨𝑡 2 𝜃(𝑡 3 ) 𝑚 2 ⟩ + ⟨𝑡 3 𝜃(𝑡 3 ) 𝑚 3 ⟩.
Together with Equation (31), this shows that

⟨𝑡 1 𝜃(𝑡 3 ) 𝑀 1 ⟩ + ⟨𝑡 2 𝜃(𝑡 3 ) 𝑀 2 ⟩ = ⟨𝑡 1 𝜃(𝑡 3 ) 𝑚 1 ⟩ + ⟨𝑡 2 𝜃(𝑡 3 ) 𝑚 2 ⟩.
We conclude the proof, using that last result with 𝜃(𝑡 3 ) = 1 and 𝜃(𝑡 3 ) = 𝑡 3 𝜒 [0,𝑠] (𝑡 3 ).

Comments, discussion

Notice that these bounds also depend on the quantities ⟨|𝑚 𝑖 |⟩ (𝑖 = 1, 2, 3).

Remark 2.

For the purpose of proving our result in the most general framework, we introduced the constants 𝜔 𝑠 and 𝜔 𝑧 , so as to determine the constants usually hidden behind the 𝒪 notation and make these constants explicit as functions depending only on 𝑠 and 𝑧 but not on ⃗ 𝑡 and 𝑅. This allows us to integrate these bounds with respect to variable ⃗ 𝑡 and obtain upper bounds for 𝛿 18 and 𝛿 23 (at the end of Sections 4.2, 4.3, respectively), therefore proving rigorous asymptotic formulas that give approximate identities more and more accurate as 𝑅 goes large.

However, the practice is usually completely different from this situation. One generally does not actually let 𝑅 tends to +∞, but one rather has some measurements on a square 𝑄 𝑅 with a given value 𝑅 and one would like to get an approximation of the moments ⟨𝑚 𝑖 ⟩ (𝑖 = 1, 2, 3) using the asymptotic formulas, together with an estimate of the error contained in the remainder. In order to obtain a small bound for the remainder, it is clearly desirable to choose 𝜔 𝑠 and 𝜔 𝑧 as small as possible, so one practically chooses 𝜔 𝑠 = 𝑠/𝑅 and 𝜔 𝑧 = 𝑧/𝑅 (as soon as 𝑅 > 𝑠). Therefore Equation (15) defines the constant 𝐶 as being equal to 𝑅, see also Remark 1.

Furthermore, instead of using the interval [-1 + (1

-𝜔 𝑠 ) 2 , 2𝜔 𝑠 + 𝜔 2 𝑠 + 𝜔 2 𝑧 ]
in Lemma 4, one can use slightly tighter intervals since all the constants are known. For instance, Equations ( 38) and (39) can be reworked to show that 𝑢 1 (𝑡 1 , 𝑡 2 , (𝑧 -𝑡 3 ), 1/𝑅) indeed ranges in the interval

[-1 + (𝑧-𝑟) 2 2𝑅 2 + (1 -𝑠 𝑅 ) 2 , 2𝑠 𝑅 + 2𝑠 2 +𝑧 2 2𝑅 2 ]
for any ⃗ 𝑡 ∈ 𝒜. Also, notice that explicit tight and rigorous constants 𝐵 (1) to 𝐵 (5) for Lemma 4 can be automatically computed on demand for a given interval using rigorous arithmetic tools such as Taylor models [START_REF] Makino | Taylor Models and Other Validated Functional Inclusion Methods[END_REF]. Together with the present article, we provide a Maple script that explicitly computes all the presented bounds, for given values 𝑠, 𝑧 and 𝑅. For the computation of the bounds of Lemma 4, we rest on a script run with the Sollya software tool [START_REF] Chevillard | Sollya: An Environment for the Development of Numerical Codes[END_REF] that provides rigorous and proven results, accounting for all possible roundoff errors in numeric computations.

The main results (4) to [START_REF] Ponomarev | Some inverse problems with partial data[END_REF] of Section 3 can be restated on the measurement area 𝑀 𝑅 ∈ {𝑄 𝑅 , 𝑆 𝑅 } as:

∫∫ 𝑀 𝑅 𝑥 𝑖 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 𝑐 (1) 0,𝑖 (𝑀 𝑅 ) ⟨𝑚 𝑖 ⟩ + 𝑐 (1) 1,𝑖 (𝑀 𝑅 ) 𝑅 (⟨𝑡 𝑖 𝑚 3 ⟩ + ⟨(𝑡 3 -𝑧) 𝑚 𝑖 ⟩) + 𝒪 ( 1 𝑅 3
)

, for 𝑖 = 1, 2 and: Actually, √ 2 appears here because it is the ratio between the perimeters of the measurements areas 𝑄 𝑅 and 𝑆 𝑅 . Similar properties appear to be true for circular shapes as well [9, Part III], and most probably also for more general geometries. Recall that 𝐵 3 [ ⃗ 𝑚, 𝑧] has vanishing mean value on R 2 by Green's theorem, which is consistent with the last equality above and with Equations ( 6) and ( 9).

∫∫ 𝑀 𝑅 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 = 𝑐 (0) 1,3 (𝑀 𝑅 ) 𝑅 ⟨𝑚 3 ⟩ + 𝑐 (0) 3,3 (𝑀 𝑅 ) 𝑅 3 ( ⟨(𝑡 2 1 + 𝑡 2 2 -2 (𝑡 3 -𝑧) 2 ) 𝑚 3 ⟩ -2 ⟨(𝑡 3 -𝑧) (𝑡 1 𝑚 1 + 𝑡 2 𝑚 2 )⟩ ) + 𝒪 ( 1 𝑅 4 ) , with 𝑐 (1) 0,𝑖 (𝑆 𝑅 ) = 𝜇 0 2 , 𝑐 (1)
This allows one to algebraically combine between the two families of expressions obtained for two different shapes in order to refine the estimates of ⟨𝑚 𝑖 ⟩ that could be obtained using a single shape. For instance, we get for 𝑖 = 1, 2:

( √ 2 -1) 𝜇 0 2 ⟨𝑚 𝑖 ⟩ = √ 2 ∫∫ 𝑄 𝑅 𝑥 𝑖 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 - ∫∫ 𝑆 𝑅 𝑥 𝑖 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 + 𝒪 ( 𝑅 ) , 2 𝜇 0 𝜋 𝑅 ⟨𝑚 3 ⟩ = 2 √ 2 ∫∫ 𝑄 𝑅 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 - ∫∫ 𝑆 𝑅 𝐵 3 [ ⃗ 𝑚, 𝑧](𝑥 1 , 𝑥 2 ) d𝑥 1 d𝑥 2 + 𝒪 ( 𝑅 )
.

Whenever 𝑅 is such that 𝑄 𝑅 and 𝑆 𝑅 are subsets of the actual measurement area, these furnish constructive approximations of ⟨𝑚 𝑖 ⟩. The precision of these estimations is driven by the size 𝑅 of the measurement area, but also by the heights 𝑠, 𝑟, 𝑧. Indeed, the error terms decrease to 0 not only when 𝑅 increases but also when 𝑠, 𝑟, 𝑧 (and 𝑟 -𝑧) decrease, as the analysis of the bounds in the preceding section and in the appendix shows.

Conclusion

We derived formulas for estimating the net moment of compactly supported magnetization distributions from measurements of the component of its associated magnetic field that is normal to the measurement plane. Those expressions can be used to essentially transform any magnetometer or technique that images the magnetic field of a magnetization distribution on a plane into a moment magnetometer. Although our main motivation was to develop a net moment estimation technique for scanning magnetic microscopy data, our formulas can be easily adapted to operate on other vector magnetic field component or (scalar) total field measurements.

A key feature in our technique is that we make minimal assumptions about source geometry and characteristics, which makes it useful to a broad range of applications. Whereas several issues of practical interest may negatively impact the quality of the moment estimates and remain to be studied in more detail, we have nevertheless provided a set of formulas that constitute the basis of net moment estimation for planar measurement geometries. Some of those issues are: (a) data discretization -magnetic data are only available at discrete points on a rectangular grid, which leads one to approximate the integrals in Equations ( 4), ( 5), ( 6), ( 7), [START_REF] Makino | Taylor Models and Other Validated Functional Inclusion Methods[END_REF], and (9) by Riemann sums, thereby making for another approximation step; (b) noisy datathe measurements themselves are often corrupted by noise whose effect on the estimates has to be systematically analyzed; (c) size of measurement area compared to magnetization support. . . For instance, it should be observed that our estimates of hm1i, hm2i are unbiased, while it is the linear combination of equations proposed in Section 6 that can gets us rid of the bias in our estimates of hm3i.

• Following Section 3: however, restricted areas of measurement, 𝑅 < 1 may be not that large BEP, work in preparation.

Appendix: proof of Lemma 1

We first establish three lemmas about Taylor expansions of order 2 that will be used for the proof of Lemma 1.

As before in Section 4, we consider two positive constants 𝜔 𝑠 and 𝜔 𝑧 such that 𝜔 𝑠 < 1 and we define 𝐶 as in Equation ( 15). In the sequel, we will extensively use the notations defined in Equations ( 16) and ( 18), together with their practical properties, as given by Equations ( 17) and ( 19).

Put

𝒟 = 𝒟(𝑠, 𝑧, 𝐶) = [-𝑠, 𝑠] 2 × (0, 𝑧] × [0, 1/𝐶]. The assumption (⃗ 𝛾, 𝜉) ∈ 𝒟 actually means ⃗ 𝛾 ∈ [-𝑠, 𝑠] 2 × (0, 𝑧], 𝜉 ∈ [0, 1/𝐶].
The following lemma explains how to compose Taylor expansions of order 2 with rigorous bounds.

Lemma 2. We consider an interval 𝐼, a function 𝑣 : 𝐼 → R and a function 𝑢 : 𝒟 → 𝐼 and we suppose that they are of the form

⎧ ⎨ ⎩ 𝑣(𝑦) = 𝑏 0 + 𝑏 1 𝑦 + 𝑏 2 𝑦 2 + 𝜀 𝑣 (𝑦) with ∀𝑦 ∈ 𝐼, |𝜀 𝑣 (𝑦)| ≤ 𝐵 3 |𝑦| 3 , 𝑢(⃗ 𝛾, 𝜉) = 𝑎 1 (⃗ 𝛾)𝜉 + 𝑎 2 (⃗ 𝛾)𝜉 2 + 𝜀 𝑢 (⃗ 𝛾, 𝜉) with ∀(⃗ 𝛾, 𝜉) ∈ 𝒟, |𝜀 𝑢 (⃗ 𝛾, 𝜉)| ≤ 𝐴 3 (𝑠, 𝑧)𝜉 3 ,
where 𝑏 0 , 𝑏

= 𝑏 0 , 𝑐 1 (⃗ 𝛾) = 𝑏 1 𝑎 1 (⃗ 𝛾), 𝑐 2 (⃗ 𝛾) = 𝑏 1 𝑎 2 (⃗ 𝛾) + 𝑏 2 𝑎 2
1 (⃗ 𝛾), and 𝐶 3 (𝑠, 𝑧) is a homogeneous polynomial of degree 3.

Proof. Let (⃗ 𝛾, 𝜉) ∈ 𝒟. From the definitions of 𝑢 and 𝑣, we get

𝜀(⃗ 𝛾, 𝜉) = 𝑏 1 𝜀 𝑢 (⃗ 𝛾, 𝜉) + 𝑏 2 ( 2𝑎 1 (⃗ 𝛾)𝜉 + 𝑎 2 (⃗ 𝛾)𝜉 2 + 𝜀 𝑢 (⃗ 𝛾, 𝜉) )( 𝑎 2 (⃗ 𝛾)𝜉 2 + 𝜀 𝑢 (⃗ 𝛾, 𝜉) ) + 𝜀 𝑣 (𝑢(⃗ 𝛾, 𝜉)).
The bound then follows from the triangle inequality, with

𝐶 3 (𝑠, 𝑧) = |𝑏 1 |𝐴 3 (𝑠, 𝑧) + |𝑏 2 | ( 2𝐴 1 (𝑠, 𝑧) + 𝐴 2,1 (𝑠, 𝑧) + 𝐴 3,2 (𝑠, 𝑧) )( 𝐴 2 (𝑠, 𝑧) + 𝐴 3,1 (𝑠, 𝑧) ) + 𝐵 3 ( 𝐴 1 (𝑠, 𝑧) + 𝐴 2,1 (𝑠, 𝑧) + 𝐴 3,2 (𝑠, 𝑧) ) 3
, and the definitions of 𝐴 1 , 𝐴 2 , 𝐴 2,1 , 𝐴 3,1 , and 𝐴 3,2 from 𝑎 1 (⃗ 𝛾), 𝑎 2 (⃗ 𝛾), and 𝐴 3 (𝑠, 𝑧), by means of Equations ( 16) and (18), together with the corresponding properties expressed in Equations ( 17) and (19).

In particular, notice that 𝑐 1 (⃗ 𝛾), 𝑐 2 (⃗ 𝛾) are homogeneous polynomials of degree 1 and 2, respectively, as in the following lemma which, in the same spirit, shows how to multiply Taylor expansions of order 2 with rigorous bounds. Lemma 3. We consider two functions 𝑢 and 𝑣 : 𝒟 → R and we suppose that they are of the form 3 and |𝜀 𝑣 (⃗ 𝛾, 𝜉)| ≤ 𝐵 3 (𝑠, 𝑧)𝜉 3 , where 𝑎 1 (⃗ 𝛾), 𝑏 1 (⃗ 𝛾), 𝑎 2 (⃗ 𝛾), 𝑏 2 (⃗ 𝛾), and 𝐴 3 (𝑠, 𝑧), 𝐵 3 (𝑠, 𝑧) are homogeneous polynomials of degree 1, 2, and 3, respectively.

⎧ ⎨ ⎩ 𝑢(⃗ 𝛾, 𝜉) = 1 + 𝑎 1 (⃗ 𝛾)𝜉 + 𝑎 2 (⃗ 𝛾)𝜉 2 + 𝜀 𝑢 (⃗ 𝛾, 𝜉) 𝑣(⃗ 𝛾, 𝜉) = 1 + 𝑏 1 (⃗ 𝛾)𝜉 + 𝑏 2 (⃗ 𝛾)𝜉 2 + 𝜀 𝑣 (⃗ 𝛾, 𝜉) with ∀(⃗ 𝛾, 𝜉) ∈ 𝒟, |𝜀 𝑢 (⃗ 𝛾, 𝜉)| ≤ 𝐴 3 (𝑠, 𝑧)𝜉
Then, we have

𝑢(⃗ 𝛾, 𝜉)𝑣(⃗ 𝛾, 𝜉) = 1 + 𝑐 1 (⃗ 𝛾)𝜉 + 𝑐 2 (⃗ 𝛾)𝜉 2 + 𝜀(⃗ 𝛾, 𝜉) with ∀(⃗ 𝛾, 𝜉) ∈ 𝒟, |𝜀(⃗ 𝛾, 𝜉)| ≤ 𝐶 3 (𝑠, 𝑧)𝜉 3 , where 𝑐 1 (⃗ 𝛾) = 𝑎 1 (⃗ 𝛾) + 𝑏 1 (⃗ 𝛾), 𝑐 2 (⃗ 𝛾) = 𝑎 1 (⃗ 𝛾)𝑏 1 (⃗ 𝛾) + 𝑎 2 (⃗ 𝛾) + 𝑏 2 (⃗ 𝛾),
+ ( 𝑎 1 (⃗ 𝛾)𝜉 + 1 2 𝑎 2 (⃗ 𝛾)𝜉 2 + 1 2 𝜀 𝑢 (⃗ 𝛾, 𝜉) )( 𝑏 2 (⃗ 𝛾)𝜉 2 + 𝜀 𝑣 (⃗ 𝛾, 𝜉) ) + ( 𝑏 1 (⃗ 𝛾)𝜉 + 1 2 𝑏 2 (⃗ 𝛾)𝜉 2 + 1 2 𝜀 𝑣 (⃗ 𝛾, 𝜉) )( 𝑎 2 (⃗ 𝛾)𝜉 2 + 𝜀 𝑢 (⃗ 𝛾, 𝜉) )
.

The bound then again follows from the triangle inequality, with

𝐶 3 (𝑠, 𝑧) = 𝐴 3 (𝑠, 𝑧) + 𝐵 3 (𝑠, 𝑧) + ( 𝐴 1 (𝑠, 𝑧) + 1 2 𝐴 2,1 (𝑠, 𝑧) + 1 2 𝐴 3,2 (𝑠, 𝑧) )( 𝐵 2 (𝑠, 𝑧) + 𝐵 3,1 (𝑠, 𝑧) ) + ( 𝐵 1 (𝑠, 𝑧) + 1 2 𝐵 2,1 (𝑠, 𝑧) + 1 2 𝐵 3,2 (𝑠, 𝑧) )( 𝐴 2 (𝑠, 𝑧) + 𝐴 3,1 (𝑠, 𝑧) )
, and the definitions of 𝐴 The following lemma collects some Taylor expansions with controlled bounds on their remainders. They will be used in the proof of Lemma 1.

Lemma 4. We define the functions 𝑣

1 to 𝑣 5 by 𝑣 1 (𝑦) = 1 √ 1+𝑦 , 𝑣 2 (𝑦) = 1 1+𝑦 , 𝑣 3 (𝑦) = √ 1 + 𝑦,
𝑣 4 (𝑦) = arctan(𝑦) and 𝑣 5 (𝑦) = arcsinh(1 + 𝑦). There exist constants 𝐵 (1) , • • • , 𝐵 (5) , depending on 𝜔 𝑠 and 𝜔 𝑧 only, such that

∀𝑦 ∈ [-1 + (1 -𝜔 𝑠 ) 2 , 2𝜔 𝑠 + 𝜔 2 𝑠 + 𝜔 2 𝑧 ], 𝑣 1 (𝑦) = 1 - 𝑦 2 + 3𝑦 2 8 + 𝜀 1 (𝑦) with |𝜀 1 (𝑦)| ≤ 𝐵 (1) |𝑦| 3 , ( 33
) 𝑣 2 (𝑦) = 1 -𝑦 + 𝑦 2 + 𝜀 2 (𝑦) with |𝜀 2 (𝑦)| ≤ 𝐵 (2) |𝑦| 3 , ( 34
) 𝑣 3 (𝑦) = 1 + 𝑦 2 - 𝑦 2 8 + 𝜀 3 (𝑦) with |𝜀 3 (𝑦)| ≤ 𝐵 (3) |𝑦| 3 , ( 35
) ∀𝑦 ∈ R, 𝑣 4 (𝑦) = 𝑦 + 𝜀 4 (𝑦) with |𝜀 4 (𝑦)| ≤ 𝐵 (4) |𝑦| 3 , ( 36
) 𝑣 5 (𝑦) = arcsinh(1) + 𝑦 √ 2 - 𝑦 2 4 √ 2 + 𝜀 5 (𝑦) with |𝜀 5 (𝑦)| ≤ 𝐵 (5) |𝑦| 3 . ( 37
)
Proof. If 𝑖 ∈ {1, 2, 3}, since 𝜔 𝑠 < 1, the function 𝑣 𝑖 is infinitely differentiable on the given interval and the existence of 𝐵 (𝑖) simply follows from Taylor's theorem at 0. If 𝑖 ∈ {4, 5}, the same argument applied on the interval [-1, 1] ensures the existence of a constant 𝐵 ′(𝑖) such that Equations (36) and (37) hold true for 𝑦 ∈ [-1, 1]. Besides, we observe that the function 𝑦 ↦ → 𝜀 𝑖 (𝑦)/𝑦 3 is continuous on (-∞, -1] ∪ [1, +∞) and tends to 0 at ±∞. Therefore, there exists a constant 𝐵 ′′(𝑖) such that Equations (36) and (37) hold true for 𝑦 ∈ (-∞, -1] ∪ [1, +∞).

Then 𝐵 (𝑖) = max{𝐵 ′(𝑖) , 𝐵 ′′(𝑖) } satisfies the requirement. Let us recall the statement that we wish to prove, namely that for any 𝑅 ≥ 𝐶 and ⃗ 𝑡 ∈ 𝒜 we ave: 3 , of degrees 3, 3, 2, and 3, respectively. Except for 𝐾 ⃗ 𝑡 (which requires some explanation), the statement is easily deduced from our previous results, using Lemma

𝐹 ⃗ 𝑡 (𝑅) = 1 √ 2 • 1 𝑅 2 + 5𝑡 1 -𝑡 2 2 √ 2 • 1 𝑅 3 + 33𝑡 2 1 -3𝑡 2 2 -6𝑡 1 𝑡 2 -10𝑡

Figure 1 :

 1 Figure 1: General view of the framework, with main notations.

Figure 2 :

 2 Figure 2: Shapes of 𝑄 𝑅 , 𝑆 𝑅 , 𝐴 𝑅 .

3 . 5 ,

 35 (3.75)] and the related formulas in [9, Part III, Sec. 3.6], taking into account the 3-D character of the slab 𝒜.

) 2 (

 2 𝑠, 𝑧) and |𝛿 4 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ Δ(4) 3 (𝑠, 𝑧) for some homogeneous polynomials Δ (𝑖) 𝑛 in the variables 𝑠 and 𝑧, of degree 𝑛. Corollary 1. Under the same hypotheses as in Lemma 1, it holds

( 5 ) 2 (

 52 𝑠, 𝑧) and |𝛿 8 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ 4Δ

( 10 ) 1 (

 101 𝑠, 𝑧) where Δ[START_REF] Weiss | Paleomagnetic analysis using SQUID microscopy[END_REF] 1 (𝑠, 𝑧) is the homogeneous polynomial of degree 1 defined by Δ[START_REF] Weiss | Paleomagnetic analysis using SQUID microscopy[END_REF]

( 3 ) 2 (

 32 𝑠, 𝑧) and |𝛿 15 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ 4Δ

( 3 ) 2 ( 4 ) 3 (

 3243 𝑠, 𝑧)𝑧|𝑚 1 ( ⃗ 𝑡)| and |𝛿 17 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ 4Δ (𝑠, 𝑧)|𝑚 3 ( ⃗ 𝑡)|. Final step. Plugging Equations (21), (22), (23) and (24) into Equation (20), we finally get

3 (

 3 𝑀 𝑅 ) = 0) .

,

  𝐿 ⃗ 𝑡 (𝑅) = -𝑣 5 (-1 + (1 -𝛾 2 𝜉) 𝑣 1 (𝑢 2 (⃗ 𝛾, 𝜉))) .

  , 𝑧), for some homogeneous polynomials Δ

  in the variables 𝑠 and 𝑧, of degree 2.

	Δ (5) 2 and Δ	(6) 2	(5)	(6) 2 (𝑠, 𝑧), for some homogeneous polynomials

, 𝑧, 𝑅) 𝑅

3

,

where |𝛿 5 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ Δ 2 (𝑠, 𝑧) and |𝛿 6 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ Δ

  From the inequalities obtained in the previous paragraphs, we see that, for any 𝑅 ≥ 𝐶, where 𝐶 is the constant given by Equation (15),

	where 𝛿 18 (𝑠, 𝑧, 𝑅) = -𝛿 |𝛿 18 (𝑠, 𝑧, 𝑅)| ≤ 𝜇 0 4𝜋 ∫∫∫ 𝒜 ( 𝜇 0 4𝜋 (	4 (	Δ (5) 2 (𝑠, 𝑧) + Δ	(3) 2 (𝑠, 𝑧)
				𝑧, 𝑅) 𝑅 3	,

9 ( ⃗ 𝑡, 𝑧, 𝑅) + 𝛿 13 ( ⃗ 𝑡, 𝑧, 𝑅) + 𝛿 16 ( ⃗ 𝑡, 𝑧, 𝑅) + 𝛿 17 ( ⃗ 𝑡, 𝑧, 𝑅) ) d ⃗ 𝑡.

  𝐶 3 (𝑠, 𝑧)𝜉3 , where 𝑐 0

1 , 𝑏 2 , 𝐵 3 ∈ R, and 𝑎 1 (⃗ 𝛾), 𝑎 2 (⃗ 𝛾), 𝐴 3 (𝑠, 𝑧) are homogeneous polynomials of degree 1, 2, 3, respectively. Then, we have (𝑣 ∘ 𝑢)(⃗ 𝛾, 𝜉) = 𝑐 0 + 𝑐 1 (⃗ 𝛾)𝜉 + 𝑐 2 (⃗ 𝛾)𝜉 2 + 𝜀(⃗ 𝛾, 𝜉) with ∀(⃗ 𝛾, 𝜉) ∈ 𝒟, |𝜀(⃗ 𝛾, 𝜉)| ≤

  and 𝐶 3 (𝑠, 𝑧) is a homogeneous polynomial of degree 3.Proof. Let (⃗ 𝛾, 𝜉) ∈ 𝒟. From the definitions of 𝑢 and 𝑣, we get𝜀(⃗ 𝛾, 𝜉) = 𝜀 𝑢 (⃗ 𝛾, 𝜉) + 𝜀 𝑣 (⃗ 𝛾, 𝜉)

  Proof of Lemma 1. Let us define 𝑢 1 (⃗ 𝛾, 𝜉) = -(𝛾 1 + 𝛾 2 )𝜉 + According to Equation (15), observe that for (⃗ 𝛾, 𝜉) ∈ 𝒟,𝛾 1 𝜉 ≤ |𝛾 1 𝜉| ≤ 𝑠𝜉 ≤ 𝑠/𝐶 ≤ 𝜔 𝑠 < 1. Therefore (1 -𝛾 1 𝜉) ≥ (1 -𝜔 𝑠 ) > 0, hence (1 -𝛾 1 𝜉) 2 ≥ (1 -𝜔 𝑠 ) 2 . Similarly, (1 -𝛾 2 𝜉) 2 ≥ (1 -𝜔 𝑠 ) 2 . 𝜔 𝑠 ) 2 ≥ -1 + (1 -𝜔 𝑠 ) 2 . (38)Besides, from the definition of 𝑢 1 , the triangle inequality and Equation (15) we obtain𝑢 1 (⃗ 𝛾, 𝜉) ≤ |𝑢 1 (⃗ 𝛾, 𝜉)| ≤ 2𝑠 𝐶 + 2𝑠 2 + 𝑧 2 2𝐶 2 ≤ 2𝜔 𝑠 + 𝜔 2 𝑠 + 𝑢 2 (⃗ 𝛾, 𝜉) = -2𝛾 1 𝜉 + (𝛾 2 1 + 𝛾 2 3 )𝜉 2 = -1 + 𝛾 2 3 𝜉 2 + (1 -𝛾 1 𝜉) 2 . 𝑡 1 ) 2 + (𝑅 -𝑡 2 ) 2 + (𝑧 -𝑡 3 ) 2 = 2𝑅 2 (1 + 𝑢 1 (⃗ 𝛾, 𝜉)), (𝑅 -𝑡 1 ) 2 + (𝑧 -𝑡 3 ) 2 = 𝑅 2 (1 + 𝑢 2 (⃗ 𝛾, 𝜉)), (𝑅 -𝑡 1 )(𝑅 -𝑡 2 ) = 𝑅 2 (1 + 𝑢 3 (⃗ 𝛾, 𝜉)),where ⃗ 𝛾 = (𝑡 1 , 𝑡 2 , (𝑧 -𝑡 3 )) and 𝜉 = 1/𝑅. Therefore, the functions 𝐹 ⃗ 𝑡 , 𝐺 ⃗ 𝑡 , 𝐾 ⃗ 𝑡 , and 𝐿 ⃗ 𝑡 can be expressed by means of 𝑣 1 to 𝑣 5 and 𝑢 1 to 𝑢 3 . Namely (for 𝐾 ⃗ 𝑡 we use the identity arctan(𝛼) = 𝛾 2 𝜉) 𝑣 2 (𝑢 2 (⃗ 𝛾, 𝜉)) 𝑣 1 (𝑢 1 (⃗ 𝛾, 𝜉)),

			𝛾 2 1 + 𝛾 2 2 + 𝛾 2 3 2	𝜉 2 = -1 +	𝛾 2 3 𝜉 2 2	+	(1 -𝛾 1 𝜉) 2 2	+	(1 -𝛾 2 𝜉) 2 2	.
	This shows that	𝑢 1 (⃗ 𝛾, 𝜉) ≥ -1 +	𝛾 2 3 𝜉 2 2	+ (1 -𝜔 2 𝑧 2	.	(39)
	We also define
	𝜋 2 -arctan(1/𝛼) which holds for 𝛼 > 0):
		𝐹 ⃗ 𝑡 (𝑅) = (1 -𝐺 ⃗ 𝑡 (𝑅) = 𝜉 2 √ 2 -𝜉 √ 2 𝑣 1 (𝑢 1 (⃗ 𝛾, 𝜉)),
		𝐾 ⃗ 𝑡 (𝑅) =	𝜋 2𝛾 3	-	1 𝛾 3	𝑣 4	(	𝛾 3 𝜉	√	2 𝑣

Reasoning as we did for 𝑢 1 , we see that for (⃗ 𝛾, 𝜉) ∈ 𝒟,

𝑢 2 (⃗ 𝛾, 𝜉) ≥ -1 + (1 -𝜔 𝑠 ) 2 , (

40

)

𝑢 2 (⃗ 𝛾, 𝜉) ≤ |𝑢 2 (⃗ 𝛾, 𝜉)| ≤ 2𝜔 𝑠 + 𝜔 2 𝑠 + 𝜔 2 𝑧 .

(41)

Finally, we define

𝑢 3 (⃗ 𝛾, 𝜉) = -(𝛾 1 +𝛾 2 )𝜉 +𝛾 1 𝛾 2 𝜉 2 = -1+(1-𝛾 1 𝜉)(1-𝛾 2 𝜉

), for which, accordingly,

𝑢 3 (⃗ 𝛾, 𝜉) ≥ -1 + (1 -𝜔 𝑠 ) 2 , (

42

)

𝑢 3 (⃗ 𝛾, 𝜉) ≤ |𝑢 3 (⃗ 𝛾, 𝜉)| ≤ 2𝜔 𝑠 + 𝜔 2 𝑠 . (

43

)

Now, for any ⃗ 𝑡 ∈ 𝒜 and any 𝑅 ≥ 𝐶, we have (𝑅 -3 (𝑢 1 (⃗ 𝛾, 𝜉)) 𝑣 2 (𝑢 3 (⃗ 𝛾, 𝜉))

  𝑧), and |𝛿 4 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ Δ

														8 √	2	2 3 + 20𝑧𝑡 3 -10𝑧 2	•	1 𝑅 4 +	𝛿 1 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 5	,
	𝐺 ⃗ 𝑡 (𝑅) =	-1 √ 2	•	1 𝑅	-	𝑡 1 + 𝑡 2 2 √ 2	•	1 𝑅 2 -	𝑡 2 1 + 𝑡 2 2 + 6𝑡 1 𝑡 2 -2𝑡 2 3 + 4𝑧𝑡 3 -2𝑧 2 8 √ 2	•	1 𝑅 3 +	𝛿 2 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 4	,
	𝐾 ⃗ 𝑡 (𝑅) =	𝜋 2(𝑧 -𝑡 3 )	-	√ 2	1 𝑅	-	√	2(𝑡 1 + 𝑡 2 ) 2	•	1 𝑅 2 +	𝛿 3 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3	,
	𝐿 ⃗ 𝑡 (𝑅) = -arcsinh(1) +	𝑡 2 -𝑡 1 √ 2	•	1 𝑅	-	3𝑡 2 1 -𝑡 2 2 -2𝑡 1 𝑡 2 -2𝑡 2 3 + 4𝑧𝑡 3 -2𝑧 2 4 √ 2	•	1 𝑅 2 +	𝛿 4 ( ⃗ 𝑡, 𝑧, 𝑅) 𝑅 3	,
	where |𝛿 1 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ Δ (1) 3 (𝑠, 𝑧),			|𝛿 2 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ Δ	(2) 3 (𝑠, 𝑧),	|𝛿 3 ( ⃗ 𝑡, 𝑧, 𝑅)| ≤ Δ	(3) 2 (𝑠,

  2 with 𝐼 = [-1 + (1 -𝜔 𝑠 ) 2 , 2𝜔 𝑠 + 𝜔 2 𝑠 + 𝜔 2 𝑧 ] for the compositions involving 𝑣 1 , 𝑣 2 , and 𝑣 3 , and 𝐼 = (-∞, +∞) for the compositions involving 𝑣 4 and 𝑣 5 , together with Lemmas 3 and 4.In the case of 𝐾 ⃗ 𝑡 a difficulty arises from the division by 𝛾 3 = 𝑧 -𝑡 3 which casts doubt on whether Δ 𝑧) can be chosen as a polynomial. Let us put 𝑑(⃗ 𝛾, 𝜉) =√ 2 𝑣 3 (𝑢 1 (⃗ 𝛾, 𝜉)) 𝑣 2 (𝑢 3 (⃗ 𝛾, 𝜉)) whence 𝐾 ⃗ 𝑡 (𝑅) = 𝜋 2𝛾 3 -1 𝛾 3 𝑣 4 (𝛾 3 𝜉 𝑑(⃗ 𝛾, 𝜉)).Thanks to our lemmas, we easily obtain that𝑑(⃗ 𝛾, 𝜉) = √ 2 + 𝑑 1 (⃗ 𝛾)𝜉 + 𝑑 2 (⃗ 𝛾)𝜉 2 + 𝜀 𝑑 (⃗ 𝛾, 𝜉),where 𝑑 1 and 𝑑 2 are homogeneous polynomials of degree 1 and 2 respectively, and |𝜀 𝑑 (⃗ 𝛾, 𝜉)| ≤ 𝐷 3 (𝑠, 𝑧)𝜉3 where 𝐷 3 is a homogeneous polynomial of degree 3. Now, from Equation (36) we see that 𝐾 ⃗ 𝑡 has the prescribed form with the remainder 𝛿 3 ( ⃗ 𝑡, 𝑧, 𝑅)/𝑅 3 being given by 𝑑 2 (⃗ 𝛾)𝜉 3 + 𝜀 𝑑 (⃗ 𝛾, 𝜉)𝜉 + 1 𝛾 3 𝜀 4 (𝛾 3 𝜉 𝑑(⃗ 𝛾, 𝜉)), with, as before, ⃗ 𝛾 = (𝑡 1 , 𝑡 2 , (𝑧 -𝑡 3 )) and 𝜉 = 1/𝑅. Now, observe that |𝑑 2 (⃗ 𝛾)𝜉 3 | ≤ 𝐷 2 (𝑠, 𝑧)𝜉 3 , |𝜀 𝑑 (⃗ 𝛾, 𝜉)𝜉| ≤ 𝐷 3,1 (𝑠, 𝑧)𝜉 3 and 𝐵 (4) |𝛾 3 | 3 |𝜉| 3 𝑑(⃗ 𝛾, 𝜉) 3 ≤ 𝐵 (4) 𝑧 2 ( √ 2 + 𝐷 1,1 + 𝐷 2,2 + 𝐷 3,3 ) 3 which proves the claim.

	(3) 2 (𝑠, ⃒ ⃒ ⃒ 1 𝛾 3 𝜀 4 (𝛾 3 𝜉 𝑑(⃗ 𝛾, 𝜉))	⃒ ⃒ ⃒ ≤	1 |𝛾 3 |

Acknowledgements. All authors were supported, in part, by an Inria grant to the associate team Impinge and by the MIT-France seed fund. The research of E. A. Lima was supported, in part, by the U.S. National Science Foundation under the grant DMS-1521765.

makes them valuable instruments for analyzing magnetized materials at fine spatial scales, e.g., in geosciences to access ancient magnetic records that might be preserved in rocks by mapping the external magnetic field generated by the magnetization within a rock sample. Recovering basic characteristics of the magnetization (such as its net moment, i.e., the integral of the magnetization over the sample's volume) is an important problem, specifically when the field is too weak or the magnetization too complex to be reliably measured by standard bulk moment magnetometers.

In this paper, we establish formulas, asymptotically exact when 𝑅 goes large, linking the integral of 𝑥 1 𝐵 3 , 𝑥 2 𝐵 3 , and 𝐵 3 over a square region of size 𝑅 to the first, second, and third


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-The creation of an "Open Science" certification for postgraduate schools -The compilation of public datasets on -"ANR requires that the full text publications resulting from projects that it finances be deposited in open access either directly in HAL or via a local institutional archive"

• And they signed "Plan S" as well as the DORA declaration…

Application: HCERES on HAL

• Plans to require that assessment campaigns be based on publications available in HAL -Communication from Michel Cosnard, January 2018

• Publishes its own assessment reports on HAL:

https://hal-hceres.archives-ouvertes.fr -OK, no licences yet, more evangelization to come -Frontiers (17%), PLOS (19%)

A

Conclusion

• Too French… ?

• The obvious advantage of coordination -Cheap and efficient management of a central publication repository -Sharing publishing experiments -Keeping our hope in public infrastructures for research

• Should we join forces?

Merci pour votre attention ! "Scientific research is a common good that we must share with everyone."

Frédérique Vidal, July 2018

  Concrete measures to re-use saved Springer budget at Université de Lorraine: • SSH: Subscription to open and public platforms (OpenEdition,France; Erudit, Canada, Open Library of Humanities, UK) • STM: Sci-Post (physics), EDP Sciences (France), Epiga (overlay journal on Episciences.org) • Joining the Fair Open Access Alliance • Support to Directory of Open Access Journals -DOAJ, Coalition of Open Access Repositories -COAR, Sparc Europe • Note: favouring free for authors-free for readers approaches

•

  ultimate PDF to XML tool "Converting PDF to XML is a bit like converting hamburgers into cows. You may be best off printing it and then scanning the result through a decent OCR package." Michael Kay (http://lists.xml.org/archives/xml-dev/200607/msg00509.html) Inspired from: Duncan Hull Sciencesconf.org -a conference management tool Services • Online conference management tool • Abstracts, full paper, peer review, conference program, registration • Over 6000 conferences so far Technologies • Modular services • Unique authentication system • Connection to HAL authorities • Upload to HAL Library support • Supervises the creation of a conference instance • Create conference series • Moderates the integration of the papers as a collection in HAL Episciences.org -an overlay journal platform Services • Overlay certification environment • Post publication peer review • "Traditional" reviewing environment • Invited editors -special issues Technologies • OAI-PMH interface with repositories • Publication: arXiv, HAL, CWI • Data: Nakala, Ortolang • Versioning • Cataloguing facilities Library support • Setting up new journals • Editorial support • Experimenting new usages: e.g. Published on 7 October 2016 • Two main articles:

•

  costs for electronic acquisitions from university libraries and research institutions,• article and book publishing charges, and • research funding for calls for projects and their beneficiaries.Application: ANR policy • Explicit reference to open access in the 2019 action plan (26 July 2018) -Abiding to the loi Pour une République numérique and the open science policy of 4 July 2018 • « l'ANR [..] demande que toutes les publications consécutives aux projets qu'elle finance, soient déposées en texte intégral dans une archive ouverte, soit directement dans HAL soit par l'intermédiaire d'une archive institutionnelle locale. » • http://www.agence-nationale-recherche.fr/PA2019

•-•

  QUICK ZOOM ON INRIA Inria -a research organisation with a vision Vision for a scientific information policy • Maximising the dissemination of our scientific assets (visibility and swift dissemination of knowledge), for a reasonable price • Constitution of a reliable and sovereign institutional corpus (documentation, preservation, access), with clear public governance principles • Contribution to shaping the scientific communication landscape in terms of editorial processes and usage made of scientific productions Deposit mandate on all scientific publications (in HAL, CC-BY) Condition to appear in annual research reports -Comprising articles in gold open access journals • Central budget for APCs -Management of a national dashboard of costs and journals • Forbidding hybrid open access • Engaging in developing new publication models -Editorial support to Episciences based journals -Investment and support to Software Heritage • Printed material as disposable goods -Creation of a central collection of reference works -On going digitization project (on HAL) Less than 10% leaks for OA payment i.e. paid directly by research teams • Very good control of hybrid OA only two leaks in 2017 • 80% come from the bio-informatics domain • Main publishers (2014-2017):

Open Science fund • Investing in open publishing systems that

  remain under the control of the scientific community • In line with the Appel de Jussieu • preprints, short-format manuscripts, data articles, open peer review, etc.

	1. Generalizing Open Access to National plan for Open Science publications
	• Issued 4th July 2018 by Frédérique Vidal, Ministre • Roadmap
	-Make open access mandatory when publishing articles and books de l'Enseignement supérieur, de la Recherche et resulting from government-funded calls for projects. de l'Innovation Strong underlying principles: -Create an
	• 50% publicly funded research "Open science seeks to create an ecosystem in which
	• Any version of the author's manuscript scientific research is more cumulative, better supported
	by data and more transparent with faster and more • Maximum of 6 month (STM), 12 months (SSH) embargo universal access to results." -Article 38: TDM • Aligning with the Open Government Partnership • For legally acquired material • We are still waiting for the implementation guidelines… (OGP)

-Support

the HAL national open repository

  Create the position of Chief Data Officer and the corresponding network within the relevant institutions.-Create the conditions for and promote the adoption of an Open Data policy for scholarly papers.

	2. Structure research data and make it
	available through Open Access
	• Roadmap
	-Make open access dissemination mandatory for research data resulting from government-funded projects.
	-
	• Additional priorities
	-Assessment based upon quality, not quantity -explicit reference to
	the San Francisco Declaration on Research Assessment (DORA)
	-Recommending open licensing for publications and data


Nicolas Rougemaille

Andreas K Schmid

Self-organization and magnetic domain microstructure of Fe nanowire arrays

published or not. The documents may come

results from the interplay between magnetocrystalline, magnetoelastic, and shape anisotropies. [START_REF]Ultrathin Magnetic Structures I[END_REF] Since the balance between the corresponding magnetic anisotropy energies might be affected by a change in the film temperature or thickness, many systems exhibit reorientations of the magnetization as a function of these parameters. [START_REF] Allenspach | [END_REF][3][4][5] Fe thin films grown epitaxially on W(110) surfaces are one interesting example of systems that show magnetization reorientation transitions.

In Fe/W(110) films with thickness in the range of 1-2 atomic monolayers, magnetic properties are complex and, under some conditions, the easy axis of magnetization can be either in-plane or perpendicular to the film plane. [6][7][8] Thicker films are magnetized within the film plane. A transition can be observed near 10 nm thickness. Below this critical thickness, the easy axis of magnetization lies in the [1-10] direction, due to a strong surface anisotropy. 9,10 Above the critical thickness, the easy axis of magnetization switches to the in-plane [001] direction, as for bulk Fe.

Here we focus on a different type of magnetic transition, which can be observed when room-temperature-grown, nanometer-thick Fe/W(110) films are annealed for a few minutes between 700 and 1000 K. During annealing the films show an in-plane magnetization reorientation from the [1-10] to the [001] direction. In this case, the magnetization reorientation was shown to be induced by the formation of coalesced three-dimensional (3D) islands elongated along the [001] direction. [11][12][13] Different factors have been proposed as the driving mechanism for the rotation of the magnetic easy axis after annealing: reduction of the magnetoelastic anisotropy and contribution of the shape anisotropy, 13 or reduction of the surface anisotropy in the coalesced 3D islands. 11 In this work, we use spin-polarized low-energy electron microscopy (SPLEEM) to obtain spatially resolved magnetic information during the morphological transition. For our study, the main advantage of SPLEEM compared to other techniques is the rapid image acquisition (typically 1 frame/s for magnetic images and video rate for topographic images) that allows us to monitor in real time the dewetting process and its effect on the magnetization reorientation. 5,14 Sample preparation and SPLEEM measurements were performed in situ under ultrahigh vacuum conditions. The Fe films were grown by molecular-beam epitaxy on a W(110) single crystal. The W substrate was cleaned by many cycles of flash heating to 2300 K under 3×10 -8 torr oxygen pressure, followed by a final temperature flash under UHV conditions. The quality of the surface was checked by Auger spectroscopy and low-energy electron diffraction. The Fe was deposited at room temperature in the microscope chamber and was annealed subsequently.

The general observation that the morphology of the islands depends on the annealing time and temperature was noted before. 13,15,16 We show here that these effects can be controlled and can be exploited to fabricate unusual self-organized structures. Our measurements show how regular arrays of Fe islands with extremely elongated shapes can be obtained and how island morphology depends on details of preparation conditions, including the substrate step structure.

We compare results from two different W single crystals: one has a higher density of atomic surface steps and, in many regions of the surface, shows pronounced step bunching. The other crystal has a substantially lower density of steps and very large, atomically flat regions. Figure 1 shows topographic images of the two surfaces. We find that the combination of moderate annealing temperatures (between 600 and 650 K) and atomically flat substrates promotes the selforganization of remarkably regular periodic arrays of Fe wires. To illustrate the formation and magnetism of Fe/W(110) nanowire arrays, we focus on 14monolayer-thick Fe films, which were grown on the two W crystals. Figures 2(a) and 2(e) show the surface topography of Fe films grown on the stepped and flat substrates, respectively, in an early stage of the dewetting process. On these images, the continuous films appear bright, while the dark regions are places where dewetting of the Fe films has occurred. In the dewetted regions, one pseudomorphic atomic Fe wetting layer remains on top of the W surface. 17 Due to the anisotropy of the W(110) substrate, voids formed in the Fe film during dewetting are preferentially elongated along the [001] direction, except on the step bunches where the high step density initiates the formation of compact-shaped voids along the step bands. With increasing annealing time, dewetted regions grow along the [001] direction, essentially forming very long, parallel grooves in the Fe film, accumulating the removed Fe along their sides. The dewetting process strongly affects the magnetization of these films, as shown in the corresponding SPLEEM images. We adjusted the spin polarization of the instrument such that in Figs. 2(b) and 2(f), bright (and dark) contrast corresponds to regions magnetized parallel (antiparallel) to the [1-10] direction, while in Figs. 2(c) and 2(g), bright and dark contrast corresponds to magnetization along the [001] direction. Note that no magnetic contrast (gray) is observed inside the grooves, since the pseudomorphic layer is magnetic only at cryogenic temperature. 6 The images shown in Figs. 2(d) and 2(h) highlight the dramatic influence of the substrate morphology: the Fe wires grown on the stepped substrate are all magnetized along the [001] direction [Fig. 2(d)], while Fe wires grown on a very flat substrate, under otherwise equivalent conditions, show a far more complex magnetic domain microstructure [Fig. 2(h)]. By observing the growth and magnetism of the wires from the start, we can attribute this difference to a subtle but important effect that takes place when the wires first nucleate. The onset of the dewetting process on flat W substrates takes place by nucleation and growth of rather elongated voids, as seen in Fig. 2(e).We observe strong magnetic contrast in Fig. 2(f) and no contrast in Fig. 2(g): this indicates that the Fe film maintains the original direction of magnetization, along [1-10], right up to the borders of the dewetted regions. The onset of dewetting is different on stepped W surfaces, where we observe the formation of chains of more compact voids in the Fe layer along pronounced substrate step bands, see Fig. 2(a). In the image shown in Fig. 2(c the [001] direction during continued annealing, the canted magnetization in the nucleation region provides a bias to initiate reorientation of the magnetization along the wire direction. As a result, Fe/W(110) wires formed on stepped crystals normally have a single-domain magnetic structure, and wire length is limited by the distance between two step bands [Fig. 2(d)]. Fe/W(110) wires grown on flat substrates can be much longer (many-micrometers length is common) and we can form wires in which magnetization remains in the [1-10] direction, perpendicular to the growth direction of the wires. With further annealing, the width of these wires continues to shrink and, as a result, we find that at some point (typically when wire width decreases below 150 nm) the magnetization of the wires flips into the [001] direction. It is likely that increasing shape anisotropy, as a result of the increasing aspect ratio, is a factor in this reorientation of the magnetization. It is interesting to note that the process of this magnetization reorientation is often accompanied with the transient formation of a multidomain magnetic structure. Figure 3 shows an example where we used the SPLEEM to map these domain structures. A topographic image of a wire is shown in (a) and bright/dark contrast in three magnetic images shows the components of the magnetization along the [1-10] (b), [001] (c), and [101] (d) directions. On the upper and lower parts of the wire, the magnetization already flipped to the [001] direction, while the middle region of the wire exhibits flux-closure domains, as can be expected in wires having a strong inplane uniaxial magnetocrystalline anisotropy, perpendicular to the wire axis. 18 The directions of magnetization in these domains, which we deduced from Figs. Using spin-polarized low-energy electron microscopy, we studied the role of the dewetting process on the magnetization reorientation mechanisms observed in annealed, nanometer-thick Fe/W(110) films. At annealing temperatures near 650 K, flat W surfaces favor a dewetting process involving the nucleation of well-defined elongated voids, which do not promote canting of the magnetization of the Fe film. In contrast, on stepped substrate surfaces we observe the formation of more compact voids. The compact voids promote in-plane canting of the magnetization in adjacent regions of the Fe layer. As the voids grow, the regions with canted magnetization provide a bias that promotes the formation of single-domain Fe wires magnetized along their growth direction. Our findings highlight how the relative importance of different magnetic anisotropy energies contributing to this magnetic reorientation can be influenced by controlling the dewetting of the Fe film.

FIG. 1 .

 1 FIG. 1. Topographic images of the stepped (a) and flat (b) W substrates. Atomic steps appear as dark lines. The dark bands in image (a) are the regions of the surface showing pronounced step bunching. The field of view is 7 µm.

  FIG. 2. Influence of the dewetting process on the magnetization reorientation. The field of view is 7 µm. Dewetted regions appear dark in the two topographic images [(a) and (e)]. The corresponding magnetic images for a spin polarization along the [1-10] [(b) and (f)] and [001] [(c) and (g)] directions show how the shape of the dewetted regions affects the magnetization distribution of the continuous film. Depending on the substrate morphology, complete annealing leads to the formation of single-domain Fe wires magnetized along the [001] direction (d), or Fe wires exhibiting more complex magnetic domain microstructure. See text for details.

  3(b)-3(d), are sketched in Fig. 3(e). Contrary to what is observed in Fe films grown on stepped W(110) substrates, the magnetization reorientation in films grown on flat surfaces involves the nucleation of domains aligned with the growth direction of the wires, resulting from the competition between the magnetocrystalline anisotropy of the Fe/W(110) layer and the shape anisotropy of the elongated Fe wires.

FIG. 3 .

 3 FIG. 3. (a) Topographic image (0.42×4.34 µm) of a wire obtained after annealing a 14-monolayerthick flat Fe/W(110) film at 600 K. Corresponding magnetic images for an incident spin polarization along the [1-10] (b), [001] (c), and [101] (d) directions. (e) Sketch of the magnetization distribution in a wire resulting from the competition between surface and shape anisotropies.

  This work was supported by the U.S. Department of Energy under Contract Nos. DE-AC02-05CH11231 and DE-AC04-94AL8500, and by the Délégation Générale pour l'Armement under Contract No. 9860830051.