The Spectral Properties of the Magnetic Polarizability Tensor for Metallic Object Characterisation

The measurement of time-harmonic perturbed field data, at a range of frequencies, is beneficial for practical metal detection where the goal is to locate and identify hidden targets. In particular, these benefits are realised when frequency dependent magnetic polarizability tensors (MPTs) are used to provide an economical characterisation of conducting permeable objects and a dictionary based classifier is employed. However, despite the advantages shown in dictionary based classifiers, the behaviour of the MPT coefficients with frequency is not properly understood. In this paper, we rigorously analyse, for the first time, the spectral properties of the coefficients of the MPT. This analysis has the potential to improve existing algorithms and design new approaches for object location and identification in metal detection. Our analysis also enables the response transient response from a conducting permeable object to be predicted for more general forms of excitation.


Introduction
In metal detection, there is considerable interest in being able to locate and identify conducting permeable objects from the measurements of mutual inductance between a transmitting and a measurement coil. Applications include security screening, archaeology excavations, ensuring food safety as well as the search for landmines and unexploded ordnance. There are also closely related topics such as magnetic induction tomography for medical imaging and eddy current testing for monitoring the corrosion of steel reinforcement in concrete structures. Within the metal detection community, magnetic polarizability tensors (MPTs) have attracted considerable interest to assist with the identification of objects when the transmitting coil is excited by a sinusoidal signal e.g [1,12,14,16,24,25,34]. Engineers believe that a rank 2 MPT provides an economical characterisation of a conducting permeable object that is invariant of position. An asymptotic formula providing the leading order term for the perturbed magnetic field due to the presence of a small conducting permeable object has been obtained by Ammari, Chen, Chen, Garnier and Volkov [5], which characterises the object in terms of a rank 4 tensor. We have shown that this simplifies for orthonormal coordinates and allows an object to be characterised by a complex symmetric rank 2 MPT, with an explicit formula for its coefficients, thus, justifying the earlier engineering conjecture [19]. We have extended Ammari et al.'s work to provide a complete asymptotic expansion for the perturbed magnetic field, which allows an object to be characterised by generalised MPTs, of which the rank 2 MPT is the simplest case [22]. In [23], we have developed asymptotic expansions for the perturbed magnetic field that describe, for a general inhomogeneous object. In the above, N 0 , R σ˚a nd I σ˚a re all real symmetric rank 2 tensors, the former describes the magnetostatic response and the latter two are frequency dependent. The frequency behaviour of the coefficients of R σ˚a nd I σ˚i s also explicitly derived. 4. The introduction of an eigenvalue problem that allows the spectral behaviour of M, and hence the frequency response of R σ˚a nd I σ˚, to be understood as a convergent infinite series using the Mittag-Leffler theorem. 5. Explicit forms of the transient response from a homogeneous conducting permeable object when the excitation is a step function or an impulse function are derived. For the step function, this rigorously shows that the long time response is that of a permeable object and, for an impulse function, the short time response is that of a perfect conductor.
The paper is organised as follows: In Section 2 some background on the characterisation of a conducting permeable object by an MPT is briefly reviewed. Then, in Section 3, a new invariant form of the MPT is presented. Section 4 describes the explicit expressions for the MPT coefficients for the limiting cases of an inhomogeneous non-conducting permeable multiply connected object at low frequency and an inhomogeneous multiply connected object with infinite conductivity (perfectly conducting). In Section 5, an energy functional is defined, from which the MPT coefficients follow, leading to explicit expressions for R σ˚a nd I σ˚. Section 6 provides bounds on R σ˚a nd I σ˚, generalising the results already known for N 0 . Then, in Section 7, explicit expressions for the eigenvalues of the tensors N 0 , R σ˚a nd I σ˚a re derived. Section 8 presents a spectral analysis of the MPT coefficients allowing their behaviour with frequency to be understood. Using this analysis, the transient response for several different forms of excitation is obtained in Section 9.

Characterisation of conducting permeable objects
We begin by considering the characterisation of a single homogenous conducting permeable object. Following [5,18], we describe a single inclusion by B α :" αB`z, which means that it can be thought of a unit-sized object B located at the origin, scaled by α and translated by z. We assume the background is non-conducting and non-permeable and introduce the position dependent conductivity and permeability as where µ 0 :" 4πˆ10´7H/m is the permeability of free space, 0 ă µ˚ă 8 and 0 ď σ˚ă 8. For metal detection, the relevant mathematical model is the eddy current approximation of Maxwell's equations since σ˚is large and the angular frequency ω " 2πf is small (see Ammari, Buffa and Nédélec [4] for a detailed justification). The electric and magnetic interaction fields, E α and H α , respectively, satisfy the curl equations in R 3 and decay as Op|x|´1q for |x| Ñ 8. In the above, J 0 is an external current source with support in B c α . In absence of an object, the background fields E 0 and H 0 satisfy (3) with α " 0. The task is to find an economical description for the perturbed magnetic field pH α´H 0 qpxq due to the presence of B α , which characterises the object's shape and material parameters by a small number of parameters separately to its location z. For x away from B α , the leading order term in an asymptotic expansion for pH α´H 0 qpxq as α Ñ 0 has been derived by Ammari et al. [5]. We have shown that this reduces to the simpler form [18,21] 1 pH α´H 0 qpxq i "pD 2 x Gpx, zqq ij pMrαBsq jk pH 0 pzqq k`p Rpxqq i " 1 4πr 3 p3r br´Iq ij pMrαBsq jk pH 0 pzqq k`p Rpxqq i .
In the above, Gpx, zq :" 1{p4π|x´z|q is the free space Laplace Green's function, r :" x´z, r " |r| andr " r{r and I is the rank 2 identity tensor. The term Rpxq quantifies the remainder and it is known that |R| ď Cα 4 }H 0 } W 2,8 pBαq . The result holds when ν P R`:" σ˚µ 0 ωα 2 " Op1q as α Ñ 0 (this includes the case of fixed σ˚, µ˚, ω as α Ñ 0). Note that (4) involves the evaluation of the background field within the object, usually at it's centre i.e. H 0 pzq, and requires it to be analytic at this location. In addition, the notation MrαBs is used to denote that M is evaluated for the configuration αB. In the following, we write M for MrαBs where no confusion arises. The rank 2 tensor M :"´C`N " p´pCq ij`p N q ij qe i b e j depends on ω, σ˚, µ˚{µ 0 , α and the shape of B, but is independent of z. This is the MPT and its coefficients can be computed from vectorial solutions θ j pξq, j " 1, 2, 3, to a transmission problem, which we will state shortly, using If the object is inhomogeneous, with possibly different piecewise-constant values of µ˚and ν in different regions of the object, then (4) and (5) still hold if we replace B with B " Ť N n"1 B pnq and B α " αB`z by B α " αB`z to describe the fact that B is made up of N regions [23]. We require that B (and B pnq ) have Lipschitz boundaries and note that µ " where 0 ă µ pnq ă 8, 0 ď σ pnq ă 8. In addition, α denotes the size of the (combined) configuration and z its location. Throughout the following, we concentrate on results for the case of B, but these readily simplify to the case of B. The aforementioned transmission problem is which is solved for θ j pξq, j " 1, 2, 3. In the above, r¨s Γ denotes the jump of the function over Γ, with Γ " BB for the homogeneous case or Γ " BB Y tBB pnq X BB pmq , n, m " 1,¨¨¨, N, n ‰ mu otherwise, and ξ is measured from an origin chosen to be in B or B, respectively.

Invariant form of M
We define Θpuq, for a constant real vector u, to be the complex vector field solution of the transmission problem where, here, and in the following, the dependence of Θpuq on position ξ is not stated explicitly for compactness of notation. In addition, Thus, it clear that θ j " Θpe j q. In addition, setting where v is also a constant real vector then, obviously, pCq ij " Cpe i , e j q, pN q ij " Npe i , e j q, and pMq ij " Mpe i , e j q are the aforementioned tensor coefficients.
To provide an alternative splitting of M, we generalise Lemma 1 of [20], which was for a homogenous object, to the inhomogeneous case, in terms of Θpuq "Θ p0q puq`Θ p1q puq´uˆξ, "Θ p0q puq`Θ p1q puq, withΘ p0q puq :" Θ p0q puq´uˆξ, as follows: Lemma 3.1. The coefficients of M in a orthonormal basis e i , i " 1, . . . , 3, can be expressed as and u, v are constant real vectors. Note that N σ˚´Cσ˚" pN σ˚p e i , e j q´C σ˚p e i , e j qqe i b e j is a complex rank 2 tensor and N 0 " N 0 pe i , e j qe i b e j is a real rank 2 tensor. The forms C σ˚, N σ˚a nd N 0 depend on the solutions Θ p0q puq, Θ p1q puq to the transmission problems and respectively, where Θ p0q puq is a real vector field and Θ p1q puq is a complex vector field.
Proof. The proof is analogous to Lemma 1 of [20].
We now consider the symmetry of the forms N 0 pu, vq, Mpu, vq and Mpu, vq´N 0 pu, vq " N σ˚p u, vq´C σ˚p u, vq and, hence, the tensors N 0 , M and M´N 0 " N σ˚´Cσ˚f or the inhomogeneous case. In the homogeneous case, the tensor N 0 can be shown to be equivalent to the Pólya-Szegö tensor parameterised by the contrast in permeability, T pµ r q, (see Lemma 3 of [20]). In addition, we have that M " N 0`O pωq " T pµ r q`Opωq as ω Ñ 0 (by Theorem 9 of [20]) and N 0 is known to be real symmetric. Consequently, the tensor N 0 provides an object characterisation for magnetostatic problems. In Lemma 4.4 of [18], we have previously shown that M is complex symmetric and provides a characterisation of homogeneous conducting permeable objects. In order to extend these results to the inhomogeneous case, for square integrable complex vector fields a, b, we will use the notation to denote the L 2 inner product over B, where the overbar denotes the complex conjugate. This reduces to a, b L 2 pBq " b, a L 2 pBq if a, b are square integrable real vector fields. Hence, }u} L 2 pBq :" u, u 1{2 L 2 pBq is the L 2 norm of u over B. We also define }u} W pc,Bq :" cu, u 1{2 L 2 pBq , for a piecewise constant c ą 0 in B, as a weighted L 2 norm of u over B. The following theorem reveals insights into N 0 pu, vq for inhomogeneous objects.
Theorem 3.2. N 0 pu, vq : R 3ˆR3 Ñ R is a symmetric bilinear form on real vectors that can be expressed as and also defines an inner product provided that µ r pξq ě 1 for ξ P B. In the above,Θ p0q puq :" Θ p0q puq´uˆξ is a real vector field, which satisfies the transmission problem Proof. We first rewrite N 0 pu, vq as where we have used Θ p0q puq "Θ p0q puq`uˆξ in (9c). The transmission problem forΘ p0q puq is also easily derived.
Corollary 3.3. It immediately follows from Theorem 3.2 that N 0 " N 0 pe i , e j qe i b e j is a symmetric tensor extending the known result for a homogenous object proved in Lemma 1 of [20]. In particular, the diagonal coefficients of the associated tensor N 0 are where the repeated index i does not imply summation. In addition, we see that N 0 ii ą 0 provided that µ r pξq ą 1 for ξ P B.
The following result provides further insights in to N 0 pu, vq when the object is homogeneous: For the homogeneous case, where B becomes B, N 0 pu, vq can also be expressed in the following alternative forms "´α where µ r " µ˚{µ 0 is now a constant.
Proof. To obtain (15a), we replace B by B and transform the volume integral over B in (9c) to a surface integral over Γ " BB and use 2u " ∇ˆpuˆξq.
As µ r is constant in B for this case, then, to obtain (15b), we subtract the following from (9c) The result then follows by transforming the remaining volume integral to a surface integral over Γ and using the transmission condition in (10).
The final result follows from integration by parts and using the far field decay conditions of Θ p0q puq.
where the repeated index i does not imply summation, and, hence, N 0 ii ą 0 if µ r ą 1 and N 0 ii ă 0 if µ r ă 1.
We now consider the symmetry of the bilinear forms Mpu, vq and Mpu, vq´N 0 pu, vq " N σ˚p u, vq´C σ˚p u, vq and, hence, the symmetry of the tensors M and M´N 0 " N σ˚´Cσ˚. Theorem 3.6. Mpu, vq : R 3ˆR3 Ñ C is a symmetric bilinear form on real vectors, which can be expressed as and N σ˚p u, vq´C σ˚p u, vq " Mpu, vq´N 0 pu, vq : R 3ˆR3 Ñ C is also a symmetric bilinear form on real vectors.
Proof. The first part of the proof applies similar arguments to Lemma 4.4 of [18], which showed that M is a symmetric tensor for a homogenous object. Here, we will apply these arguments to the form Mpu, vq and consider an object with possibly inhomogeneous materials. We begin by noting from (8a) that by use of the transmission problem (7). Next, by integration by parts, we have ż since rnˆΘpvqs BB pnq YBB pmq " 0 for n, m " 1, . . . , N, n ‰ m. It then follows that ż B ∇ˆµ´1 r ∇ˆΘpuq¨Θpuqdξ " From the above, and Mpu, vq " Npu, vq´Cpu, vq, the result in (16) immediately follows. On consideration of (16), and the linearity of the transmission problem (11), we see that Mpu, vq " Mpv, uq @u, v P R 3 , Mpu`w, vq " Mpu, vq`Mpw, vq @u, v, w P R 3 and Mpcu, dvq " cdMpu, vq @u, v P R 3 and @c, d P R and, thus, Mpu, vq : R 3ˆR3 Ñ C is a symmetric bilinear form on real vectors. By using Theorem 3.2, it follows that Mpu, vq´N 0 pu, vq : R 3ˆR3 Ñ C is also a symmetric bilinear form on real vectors. Corollary 3.7. It immediately follows from Theorem 3.6 that M " Mpe i , e j qe i be j is a complex symmetric tensor and M´N 0 " N σ˚´Cσ˚" pN σ˚p e i , e j q´C σ˚p e i , e j qqe i b e j is a complex symmetric tensor, extending the known results in Lemma 4.4 of [18] and Lemma 1 of [20] for a homogeneous object to the inhomogeneous case.
Remark 3.8. Note that the first and last terms in (16) cannot be expressed in terms of the notation introduced in (12) since Θpuq and Θpvq are complex valued and the integrands each lack a complex conjugate.

Limiting cases of M
Recall that the asymptotic formula (4) is valid for ν " Op1q as α Ñ 0 and so care needs to be exercised when interpreting the limiting cases of M. Still further, recall that the eddy current model (2) is a low-frequency approximation of the Maxwell system and so the limit of ν Ñ 8 for fixed σ˚, α would break break both (4) and (2). The case of a perfect conductor with sufficiently small ω, α and σ˚Ñ 8 is permitted by the eddy current model, provided topological requirements on B are satisfied [4,31], but invalidates (4) as we still have ν Ñ 8. In the following, we compute Mpu, v; νq when ν " 0 and ν Ñ 8. From the former, we can deduce Mp0q, which provides a magnetostatic characterisation of B for a permeable object, and, from the latter, we can obtain Mp8q, which we denote as the characterisation of a perfectly conducting object. The coefficients of Mp8q can not be substituted in to (4) and, instead, should be viewed as the limiting characterisation of B provided by (4) as σ˚Ñ 8 and α Ñ 0.
whereΘ p0q puq is the real vector field solution to (14) and Θ p8q puq is the real vector field solution Proof. Using Lemma 3.1, we immediately establish that Cpu, vq vanishes for ν " 0 and, from (11), find that Θ p1q puq " 0 for ν " 0. Thus, Mpu, v; 0q " N 0 pu, vq and we quote the form of N 0 pu, vq given in Theorem 3.2.
For an object with homogeneous µ˚, Mp0q " Mpe i , e j , 0qe i b e j " N 0 " N 0 pe i , e j qe i b e j is just the Póyla-Szegö tensor parameterised by the contrast in permeability T pµ r q, independent of the object's topology.
Proof. The result follows from Lemma 4.1 and by applying similar arguments to Lemma 3 of [20]. The latter discusses the contractibility of loops associated with holes in the object so that the result is independent of the object's topology.
In the case where B becomes a single object B, with Betti numbers such that β 1 pBq " β 1 pB c q " 0, then pMp8qq ij " Mpe i , e j ; 8q can be expressed as and coincides with the coefficients of the Póyla-Szegö tensor parameterised by 0, pT p0qq ij . In the above, δ ij denotes the Kronecker delta and ψ i pξq solves Proof. For the case of β 1 pBq " β 1 pB c q " 0, it follows from Lemma 4.1 that we can set ∇Θ p8q pe i q "´2∇ψ i where ψ i solves (21). Also, by applying integration by parts, the different forms of pMp8qq ij " Mpe i , e j ; 8q in (20) can be easily obtained. We see this coincides with pT p0qq ij by comparing (20c) to the expression for T pcq given in (9) of [20] in the case where the contrast becomes 0.
Remark 4.4. For objects, with β 1 pB c q ‰ 1 then ∇ˆΘ p8q pe i q "´2∇ψ i`hi in B c where h i is a curl free function that is not a gradient with dimension dimph i q " β 1 pB c q. Unlike in Lemma 3 of [20], the loops γ k pB c q, k " 1, . . . , β 1 pB c q associated with the holes passing through the object are no longer contractable and so h i ‰ 0 in this case. Thus, pMp8qq ij does not coincide with pT p0qq ij for objects with holes and the more general form pMp8qq ij " Mpe i , e j ; 8q following from (18) must be used. Numerical examples illustrating this for single multiply connected objects with loops were presented in [20]. 5 The energy functional associated with M An important alternative representation of Mpu, vq´N 0 pu, vq " N σ˚p u, vq´C σ˚p u, vq is provided in the following theorem.
Theorem 5.1. The bilinear form Mpu, vq´N 0 pu, vq " N σ˚p u, vq´C σ˚p u, vq can be written as R σ˚p u, vq`iI σ˚p u, vq where R σ˚p u, vq : R 3ˆR3 Ñ R, I σ˚p u, vq : R 3ˆR3 Ñ R are the following symmetric bilinear forms on real vectors Additionally,´R σ˚p u, vq and I σ˚p u, vq define inner products on real vectors.
Proof. Using the definitions in (9a) and (9b) and the transmission problem (11) we have Then, using uˆξ " 1 iν ∇ˆµ´1 r ∇ˆΘ p1q puq´pΘ p0q puq`Θ p1q puq´uˆξq in B we have, for u P R 3 , Denoting the latter two terms by´α 3 4 A 1 and α 3 2 A 2 , respectively, then, by integration by parts, we have Next, using ∇ˆµ´1 r ∇ˆΘ p0q puq " 0 in B, integrating by parts the third integral over B, and expanding the integral over B c , we have Note that the final equality follows by cancelling terms and using ∇ˆµ´1 r ∇ˆΘ p0q puq " 0 in B c . In addition, by transforming the surface integral in A 2 , we have Denoting the real part of pN σ˚´C σ˚q pu, vq as R σ˚p u, vq and its imaginary part by I σ˚p u, vq then we have By the properties of the complex conjugate, we get that where, in the final step, we have used which follows since we know that R σ˚p u, vq " R σ˚p v, uq by the symmetry of N σ˚p u, vqĆ σ˚p u, vq in Theorem 3.6 and, hence, the symmetry of its real and imaginary parts. By applying similar arguments to I σ˚w e get that and Still further, using (26), (24) becomes (22a) and, in a similar manner, using (25), (27) becomes (22b) as desired. It also follows from (22a), and the linearity of the transmission problem (11), that R σ˚p u, vq " R σ˚p v, uq @u, v P R 3 , R σ˚p u`w, vq " R σ˚p u, vq`R σ˚p w, vq @u, v, w P R 3 and R σ˚p cu, dvq " cdR σ˚p u, vq @u, v P R 3 and @c, d P R and, thus, R σ˚p u, vq : R 3ˆR3 Ñ R is a symmetric bilinear form on real vectors. Similarly, I σ˚p u, vq : R 3ˆR3 Ñ R is a symmetric bilinear form on real vectors. In addition, since´R σ˚p u, uq ě 0, I σ˚p u, uq ě 0 and R σ˚p u, uq " I σ˚p u, uq " 0 only if u " 0,´R σ˚p u, vq and I σ˚p u, vq define inner products on real vectors.

Corollary 5.2. An alternative splitting of the MPT is
R σ˚p e i , e j qe i b e j and I σ˚" I σ˚p e i , e j qe i b e j are real symmetric tensors. In addition, pN 0 q ii ě 0, pR σ˚q ii ď 0 and pI σ˚q ii ě 0 where the repeated index i does not imply summation.
Proof. The splitting M " N 0`Rσ˚`i I σ˚i mmediately follows from Theorem 5.1. The symmetry of N 0 " N 0 pe i , e j qe i be j follows from (13) and the symmetries of R σ˚" R σ˚p e i , e j qe i be j and I σ˚" I σ˚p e i , e j qe i b e j from (22a) and (22b), respectively. The diagonal coefficients of N 0 are quoted in Corollary 3.3 and those of R σ˚a nd I σ˚a re leading to the quoted result.

Remark 5.3. In Theorems 3.2 and 5.1 we have established that the MPT follows from the symmetric bilinear form
where u and v are real vectors and N 0 pu, vq,´R σ˚p u, vq and I σ˚p u, vq are symmetric bilinear forms and inner products. This suggests that another possible route to the derivation of the asymptotic formula for pH α´H 0 qpxq could be through through the approach of topological derivatives [28], where, through the definition of an appropriate energy functional, its topological derivative is the leading order term of (4). Still further, N 0 pu, vq defines a magnetostatic type energy,´R σ˚p u, vq a magnetic type energy and I σ˚p u, vq an electric (Ohmic) type energy functional for pairs of solutions Θpuq and Θpvq, which provides a concrete interpretation of the three contributions in (29).
We complete this section by establishing an alternative form of R σ˚p u, vq and I σ˚p u, vq. To do this, we first remark that the weak form for the transmission problem (11) is: Find Θ p1q puq P X such that where X :" tϕ P Hpcurlq : ∇¨ϕ " 0 in B c , ϕ " Op|ξ|´1q as |ξ| Ñ 8u.
We can then establish the following result: Lemma 5.4. An alternative form of the symmetric bilinear forms R σ˚p u, vq and I σ˚p u, vq, introduced in (22a) and (22b), respectively, is , and, hence, from (22a) we obtain that which must be real by definition. Also, by using the transmission problem (11) and recalling Θ p0q puq P R 3 , we have that which must be real by definition. Then, since R σ˚p u, vq P R, Θ p0q puq P R 3 and Θ p1q puq P C 3 , it follows that the first term is purely imaginary, the second is complex and the last term is real and, hence, an alternative form of I σ˚p u, vq is given by (31b). Still further, we have and, from this, we immediately obtain (31a). 6 Bounds on the off-diagonal coefficients of R σ˚a nd I σB ounds on the off-diagonal coefficients of the Pólya-Szegö tensor, and hence N 0 for homogenous µ˚, have previously been established e.g. [7,17,20]. The following provides a bound on the magnitudes of the off-diagonal coefficients of R σ˚a nd I σ˚.
Proof. First we construct an upper bound on |pR σ˚q ij | for i ‰ j as which follows by application of the Cauchy-Schwartz inequality. From pc´dq 2`2 cd " c 2`d2 we have cd ă 2cd ă c 2`d2 for real c ą 0 and d ą 0 and so as desired. In a similar fashion, for i ‰ j, since pI σ˚q kk ą 0 giving the result as desired.
7 Eigenvalues of R σ˚, I σ˚a nd N 0 As R σ˚a nd I σ˚a re real symmetric tensors, their coefficients, when arranged in the form of a 3ˆ3 matrices, can be diagonalised by orthogonal matrices Q R σ˚a nd Q I σ˚, respectively, so that Λ R σ˚a nd Λ I σ˚a re diagonal and pΛ R σ˚q ij "ppQ R σ˚q T R σ˚QR σ˚q ij " pQ R σ˚q ki pR σ˚q kp pQ R σ˚q pj , (33a) pΛ I σ˚q ij "ppQ I σ˚q T I σ˚QI σ˚q ij " pQ I σ˚q ki pI σ˚q kp pQ I σ˚q pj .
Moreover, the diagonal entries of Λ R σ˚a nd Λ I σ˚a re the eigenvalues of R σ˚a nd I σ˚, respectively, and the columns of the matrices Q R σ˚a nd Q I σ˚a re their eigenvectors. In a similar way, N 0 can be diagonalised by the orthogonal matrix Q N 0 containing the eigenvectors of N 0 so that are the elements of a diagonal matrix containing the eigenvalues of N 0 . The orthogonal matrices Q R σ˚, Q I σ˚a nd Q N 0 can also be viewed as rotations of the object B such that R σ˚r Q R σ˚p Bqs, I σ˚r Q I σ˚p Bqs and N 0 rQ N 0 pBqs 2 are diagonal and their entries being the associated eigenvalues. We summarise this as the main result of this section: Theorem 7.1. The eigenvalues of R σ˚, I σ˚a nd N 0 can be explicitly expressed as the diagonal coefficients where the repeated index i does not imply summation and Θ p1q puq is the solution to with Q " Q R σ˚, Q " Q I σ˚, respectively. In addition, Θ p0q puq "Θ p0q puq`uˆξ is the solution Proof. Under the action of a rotation Q, the MPT's coefficients transform as pM 1 q ij " pMrQpBqsq ij " pQq ip pQq jq pMq pq . Thus, pN 0 1 q ij`p Q σ 1 q ij`i pI σ 1 q ij "pQq ip pQq jq ppN 0 q pq`p R σ˚q pq`i pI σ˚q pq q pN 0 1`R σ 1 q ij`i pI σ 1 q ij "pQq ip pQq jq pN 0`Rσ˚q pq`i pQq ip pQq jq pI σ˚q pq , and so pN 0 1 q ij " pN 0 rQpBqsq ij " pQq ip pQq jq pN 0 q pq , pR σ 1 q ij " pR σ˚r QpBqsq ij " pQq ip pQq jq pR σ˚q pq and pI σ 1 q ij " pI σ˚r QpBqsq ij " pQq ip pQq jq pI σ˚q pq . Choosing Q " Q R σ˚, and noting that under the action of this rotation B becomes Q R σ˚p Bq, then, by the application of Corollary 5.2 for the rotated object configuration we have for the diagonal coefficients, where the repeated index i does not imply summation. Repeating similar steps for Q " Q I σ˚a nd Q " Q N 0 gives the corresponding result for Λ I σ˚a nd Λ N 0 .
Remark 7.2. When Q " Q R σ˚i s applied to B, the resulting R σ 1 " R σ˚r Q R σ˚p Bqs will necessarily be diagonal and will have the eigenvalues of R σ˚r Bs as its diagonal coefficients, i.e. Λ R σ˚" R σ 1 . Since R σ 1 is diagonal, the eigenvalues of R σ 1 are its diagonal entries, i.e. Λ R σ 1 " R σ 1 , and the eigenvectors of R σ 1 form the columns of I. However, the eigenvectors of R σ˚r Bs do not, in general, form the columns of I unless the object has rotational or reflectional symmetries. It follows that the eigenvalues contained in Λ R σ˚" Λ R σ 1 are invariant under the action of rotation of an object, but the eigenvectors of R σ˚a re not. Using similar, arguments we also get that Λ I σ 1 " Proof. Choosing ν P p0, 8q excludes the limiting cases of zero frequency and infinite conductivity [20]. The definiteness of R σ˚, I σ˚, for ν P p0, 8q, and N 0 , for µpξq ą 1 for ξ P B, follow from Theorem 7.1. The results on N 0 for a homogeneous object follow from [7, pg. 93], since N 0 coincides with the Pólya-Szegö tensor for a homogenous object. The results on the limiting cases follow from (17) and (18) by considering ν " 0 and ν Ñ 8, respectively.
8 Spectral analysis of M " N 0`Rσ˚`i I σ˚f or an object with homogeneous σI n this section, we investigate how M depends on ω. An illustration of the typical behaviour of RepMpωqq " N 0`Rσ˚p ωq and ImpMpωqq " I σ˚p ωq for the case of a conducting sphere B α " αB with radius α " 0.01m and material parameters µ r " 1.5 and σ˚" 5.96ˆ10 6 S/m is shown in Figure 1. This plot is obtained by evaluating the known analytical solution provided by Wait [32]. Here, Λ R σ˚, Λ I σ˚a nd Λ N 0 each contain a single repeated eigenvalue of multiplicity three as R σ˚, N 0 and I σ˚p ωq are each a multiple of I. Numerical results for other object shapes can be found in [20,21,23]. The matrices of eigenvalues Λ R σ˚a nd Λ I σ˚a re strongly dependent on ν " α 2 σ˚ωµ 0 . The limiting behaviour of M´N 0 " R σ˚`i I σ˚"´q C σ˚`N σ˚f or ν " 0 and ν Ñ 8 has already been investigated and we recall that • R σ˚Ñ 0, I σ˚Ñ 0 as ν Ñ 0 and hence Λ R σ˚Ñ 0, Λ I σ˚Ñ 0 as ν Ñ 0; • I σ˚Ñ 0 as ν Ñ 8 and hence since N 0`Rσ˚Ñ Mp8q as ν Ñ 8 by Lemma 4.1 then Λ I σ˚Ñ 0 and Λ N 0`Rσ˚Ñ Λ Mp8q as ν Ñ 8. If β 1 pBq " 0 then Mp8q simplifies to T p0q and Λ N 0`Rσ˚Ñ Λ T p0q . Throughout this section we require that σ˚" σ pnq is constant throughout B, but allow µ˚to still vary in a piecewise constant manner through B. With this, and the above in mind, it is beneficial to consider the dependence of M, R σ˚a nd I σ˚o n ν from which their behaviour with ω can be readily obtained by a simple change of variables. As explained previously in Section 4, our interest lies in the case in understanding the behaviour of these tensors where ν " Op1q so as not to invalidate (4). We begin by investigating the behaviour of Θ p1q puq with ν.

Spectral behaviour of Θ p1q puq with ν
We introduce the model eigenvalue problem: Find the eigenvalue-eigensolution pairs pλ, φq such that which we will show is closely related to understanding the behaviour of Θ p1q puq. The model eigenvalue problem can be written in weak form as: Find φ P Y and λ such that where Y :"tϕ P Hpcurlq : To analyse (39), it is useful to apply a Helmholtz decomposition [26, pg. 86] to Y : for ξ P B. Corresponding to the eigenvalue λ " 0, then it can be shown that φ 0 " 0 and there are an infinite number of gradient eigenfunctions in B. Corresponding to λ ‰ 0 the problem (39) can be rewritten as: Find φ 0 P Y 0 and λ ‰ 0 such that Choosing ψ " φ 0 in (41) it is possible to show that λ ą 0 . Continuing to follow Monk, then, by introducing an appropriate solution operator, the existence of eigenvalues and eigenfunctions can be established using the Hilbert-Schmidt theory leading to the following conclusions: 1. Corresponding to the eigenvalue λ " 0 there is an infinite family of eigenfunctions, which are such that φ " ∇p in B for any p P S.
For ν P p0, 8q, it is clear that iν is not an eigenvalue of (39) and, hence, by Corollary 4.19 in Monk [26, pg. 98] the problem: Find Θ p1q puq P Y such that has a unique solution for every Θ p0q puq P L 2 pBq Ă Y 1 . Defining the operator pL´iνIq : Y Ñ Y 1 , this problem consists of finding the solution to the operator equation Writing A " pL´iνIq´1 : Y 1 Ñ Y for the solution operator defined by µ´1 r ∇ˆpAf q, ∇ˆg L 2 pBq´i ν Af , g L 2 pBq " f , g L 2 pBq @g P Y, then it clear that A is linear and we can check that it is self adjoint: f , Ag L 2 pBq " µ´1 r ∇ˆpAf q, ∇ˆpAgq L 2 pBq´i ν Af , Ag L 2 pBq " µ´1 r ∇ˆpAgq, ∇ˆpAf q L 2 pBq´i ν Ag, Af L 2 pBq " g, Af L 2 pBq " g, Af L 2 pBq " Af , g L 2 pBq .
Also, using the spectral behaviour of L from Remark 8.1, we have Lφ n " λ n φ n , thus, pLí νIqφ n " pλ n´i νqφ n and, hence, Aφ n " pλ n´i νq´1φ n . Furthermore, as A is linear and self adjoint, the spectral theorem applies to A, which, when combined with (44), leads immediately to (42). We can extend its applicability to ν P r0, 8q since we know that Θ p1q puq vanishes for ν " 0. Then, we introduce β n :"´iν{piν´λ n q and its real and imaginary parts are trivially computed.
This estimate goes to zero as n Ñ 8 and, hence, (42) converges.

Corollary 8.3. From the definition of β n in Lemma 8.2 it follows that
and d 2 dplog νq 2 pRepβ n qq "´4 Remark 8.4. The complex functions β n pνq, n " 1, 2, . . ., characterise the behaviour of Θ p1q puq with respect to ν. The real part of each function, Repβ n q, is monotonic and bounded with log ν and the imaginary part of each function, Impβ n q, has a single local maximum with log ν.

Spectral behaviour of R σ˚a nd I σ˚w ith ν
The following Lemma, which describes the behaviour of R σ˚a nd I σ˚w ith ν, follows from the representation of Θ p1q puq provided by Lemma 8.2.
Lemma 8.5. The coefficients of the tensors R σ˚a nd I σ˚f or an object with homogeneous σ˚, although not necessarily homogenous µ˚, can be expressed as the convergent series pR σ˚q ij "´α Repβ n qλ n φ n , Θ p0q pe i q Impβ n qλ n φ n , Θ p0q pe i q Proof. Using Theorem 5.1 and Lemma 8.2 we see that pR σ˚q ij can be expressed as Then, noting that µ´1 r ∇ˆP n pΘ p0q pe i qq, ∇ˆP m pΘ p0q pe j qq and combining with (49), gives the desired result for pR σ˚q ij . For pI σ˚q ij we have , and using The final result for pI σ˚q ij follows from noting that φ n , φ m L 2 pBq " δ mn .
The convergence of (48a) and (48b) follows in a similar manner to that of (42) by usinǧˇˇˇ as n Ñ 8 with s ą 2 and C ą 0 independent of λ n .
Taking in to account possible multiplicities in the eigenvalues λ n , we have the following: Remark 8.6. The result of Lemma 8.5 can be rewritten to make explicit possible multiplicities in the eigenvalues λ n as pR σ˚q ij " We observe that Lemma 8.5 provides a connection between the point of inflection of pR σ˚q ij with log ν and the stationary point of pI σ˚q ij with log ν as discussed in the following remark.
Remark 8.7. Applying (47) to the results (51) then a point of inflection for pR σ˚q ij with log ν corresponds to where Similarly, the stationary point for pI σ˚q ij with log ν corresponds to where d dplog νq ppI σ˚q ij q " Thus, a stationary point for pI σ˚q ij with respect to log ν corresponds to a point of inflection for pR σ˚q ij with respect to log ν.
8.2.1 Dominant spectral behaviour of pR σ˚p νqq ij , pI σ˚p νqq ij From Corollary 8.3, we observe, for i " j, that the expressions (51a) and (51b) involve sums of terms that are each monotonically decreasing and bounded with log ν and have a single local maximum with log ν, respectively. For i ‰ j, (51a) involves sums of terms that are either monotonically decreasing and bounded or monotonically increasing and bounded with log ν, and, (51b) has terms which have either a single local minimum or maximum with log ν. The difference in the behaviour of the different terms for i ‰ j is due to , whose sign can vary for different n. For each i, j we expect, amongst the terms in these summations, there is a n " n dom , which we call the dominant mode, that provides the dominant behaviour of pR σ˚p νqq ij and pI σ˚p νqq ij for ν P r0, ν max q. We confirm this behaviour by using a least squares fit of the functions to the curves of pR σ˚p νqq ij " pRepMpνqqq ij´p N 0 q ij and pI σ˚p νqq ij " pImpMpνqqq ij where a and c control the amplitude and sign of the functions and we expect to find that b « d corresponds to the dominant eigenvalue λ n dom for the considered coefficient. First, we consider the conducting sphere previously shown in Figure 1. For this object, R σ˚p νq and I σ˚p νq are diagonal and a multiple of I. We also expect the dominant mode to be n dom " 1, which has an eigenvalue with multiplicity 3. By fitting the functions f pR σ˚q ii pa, bq and f pI σ˚q ii pc, dq to the exact data (no summation implied) for f P r0, 10 4 qHz, where f max " ω max {p2πq " 10 4 Hz, which implies ν max " α 2 σ˚µ 0 ω max « 47, we find b « d « 10, as expected. In Figure 2, we observe that the functions provide a good approximation of pR σ˚p νqq ii and pI σ˚p νqq ii . Also included are the residuals´|pR σ˚p νqq ii´f pR σ˚q ii pa, bq| and |pI σ˚p νqq ii´f pI σ˚q ii pa, bq|, which are small for ν P r0, ν max q. As a second example, we consider an irregular conducting tetrahedron B α " αB where the object B has vertices p0, 0, 0q, p0.7, 0, 0q, p0.89, 0.46, 0q and p1.36, 1.33, 1.62q, α " 0.01m, µ r " 1.5 and σ˚" 5.96ˆ10 6 S/m. For this object, R σ˚p νq and I σ˚p νq have 6 independent coefficients and, therefore, for each coefficient, the dominant mode may differ. The functions f pR σ˚q ij pa, bq and f pI σ˚q ij pc, dq are fitted to the curves pR σ˚p νqq ij and pI σ˚p νqq ij obtained using the computational procedure described in [18,20] for f P r0, 10 5 qHz using a mesh of 34 473 unstructured tetrahedra and third order finite elements. Different values of c « d are obtained for each coefficient and we observe, in Figure 3, for the diagonal coefficients, and in Figure 4, for the off-diagonal coefficients, that the functions describe the dominant behaviour of pR σ˚p νqq ij and pI σ˚p νqq ij for f P r0, 10 5 qHz, where f max " ω max {p2πq " 10 5 Hz, which implies ν max " α 2 σ˚µ 0 ω max « 470. The presence of dominant modes also provides further insights in to how pR σ˚q ii and pI σ˚q ii are connected as described in the following remark.
where C ą 0 depends on λ n dom , φ n dom ,k and Θ p0q , but is independent of ν, which reveals insights in to howˇˇd d log ν ppR σ˚q ii qˇˇand pI σ˚q2 ii are connected. From frequency sweeps of the computed tensor coefficients for different objects with homogenous σ˚(e.g. [20,21,23] ), and from broadband measurements of tensorial coefficients (e.g. [30,13,27]), pR σ˚q ii has been found to exhibit a monotonic and bounded behaviour with log ν and I σi i has a single local maximum with log ν for a large range of objects. Thus, one might be tempted to conjecture thaťˇˇˇd d log ν ppR σ˚q ii qˇˇˇˇď CpI σ˚q ii , however, this is not true, the correct behaviour being of the type stated in (52).  Figure 3: Conducting irregular tetrahedron with α " 0.01m, µ r " 1.5 and σ˚" 5.96ˆ10 6 S/m: Curve fitting of paq R σi i pνq " RepMpνqq ii´N 0 ii and pbq I σi i pνq " ImpMpνqq ii , i " 1, 2, 3 (no summation implied).

Reduction in the number of coefficients in R σ˚p νq, I σ˚p νq due to object symmetries
The important role played by φ n,k , Θ p0q pe i q L 2 pBq φ n,k , Θ p0q pe j q L 2 pBq in the transformation of R σ˚a nd I σ˚i s understood through the following lemma. Lemma 8.9. Under the action of an orthogonal transformation matrix Q φ n,k , Θ p0q pe i q L 2 pQpBqq φ n,k , Θ p0q pe j q L 2 pQpBqq " pQq ip pQq jq φ n,k , Θ p0q pe p q L 2 pBq φ n,k , Θ p0q pe q q transforms like the coefficients of a rank 2 tensor. Consequently, the coefficients of R σ˚a nd I σ2  Figure 4: Conducting irregular tetrahedron with α " 0.01m, µ r " 1.5 and σ˚" 5.96ˆ10 6 S/m: Curve fitting of paq pR σ˚p νqq ij " pRepMpνqqq ij´p N 0 q ij and pbq pI σ˚p νqq ij " pImpMpνqqq ij , i ‰ j .
Proof. Using the notation Θ p0q B puq to denote the solution of (10) and φ n,k,B to denote the n, k eigenmode of (38), where the dependence on B has been made explicit, we have, from Proposition 4.3 of [6], the transformations Θ p0q QpBq puq " |Q|QΘ p0q B pQ T uq, φ n,k,QpBq " |Q|Qφ n,k,B , for an orthogonal transformation matrix Q. Observe that φ n,k,RpBq does not depend on auxiliary vector and so its transformation is simpler. Following similar arguments to the proof of Theorem 3.1 of [18] we have where we have used Θ p0q B pQ T e i q " B pe p q in the final step. Repeating similar steps for φ n,k , Θ p0q pe j q L 2 pQpBqq gives the result in (53). On consideration of (51) the transformations of the coefficients of R σ˚r QpBqs and I σ˚r QpBqs immediately follow.
Remark 8.10. Suppose, due to reflectional or rotational symmetries of an object, that pR σ˚q ij " 0 and pI σ˚q ij " 0 for some i ‰ j. According to Lemma 8.9, we have already seen φ n,k , Θ p0q pe i q L 2 pBq φ n,k , Θ p0q pe j q L 2 pBq transforms like the coefficients of a rank 2 tensor. This then implies must hold for each n to ensure that (51) results in pR σ˚q ij " 0 and pI σ˚q ij " 0, independent of the object's materials and the frequency. It is impossible to have φ n,k , Θ p0q pe i q L 2 pBq " 0 or φ n,k , Θ p0q pe j q L 2 pBq " 0 for all k since this would then imply that all of the ith row or the jth column of the tensor was 0, which contradicts Lemma 4.1 where the diagonal coefficients only go to 0 for extreme values. Furthermore, this also implies that if we have a rotational, or reflectional symmetries resulting in pR σ˚q ij " 0 and pI σ˚q ij " 0 for some i ‰ j, then we must also have multpλ n q ě 2 for all n.
To understand how the eigenvectors of R σ˚a nd I σ˚f or a general object can differ, we consider the following Lemma that provides estimates on the off-diagonal elements of the commutators using the alternative form of the tensors provided by Lemma 5.4. We note that it is easy to show that the diagonal elements of the commutators, corresponding to i " k in (55), always vanish for any object.
Remark 8.12. In [23] we have proposed to use the eigenvalues of R σ˚a nd I σ˚f or the classification of objects, as they are known to be invariant under an object rotation, and their eigenvectors for determining an object's orientation. Lemma 8.11 shows that the off-diagonal elements of the commutator between R σ˚p νq and I σ˚p νq, for general objects, grows at most linearly with ν. Recalling that ν " ωσ˚µα 2 , then, by using d dω |pR σ˚p ωqq ij pI σ˚p ωqq jk´p I σ˚p ωqq ij pR σ˚p ωqq jk | , over a range of ω, will also provide useful information and allow cases where the eigenvectors of R σ˚p ωq and I σ˚p ωq are the same and where they differ to be distinguished. As an illustration, we include, in Figure 5, the numerical results for |pZpωqq ik | " |pR σ˚p ωqq ij pI σ˚p ωqq jkṕ I σ˚p ωqq ij pR σ˚p ωqq jk |, i ‰ k, for the irregular tetrahedron previously considered in Figures 3  and 4. We observe that the behaviour of |pZpωqq ik | tracks }R σ˚} F }I σ˚} F and this behaviour is similar, in turn, to the estimate in (61).

Mittag-Leffler expansion of M
Given a meromorphic function f pwq in a region Ω with poles a n , then Ahlfors [2, pg. 187] explains how it can be expressed in the form where P n p1{pw´a n qq is a polynomial in 1{pw´a n q for each pole a n and gpwq is analytic in Ω. Unfortunately, the sum on the right hand side is infinite and so there is no garuntee that it   Figure 5: Conducting irregular tetrahedron with α " 0.01m, µ r " 1.5 and σ˚" 5.96ˆ10 6 S/m: Behaviour of |pZpωqq ik | " |pR σ˚p ωqq ij pI σ˚p ωqq jk´p I σ˚p ωqq ij pR σ˚p ωqq jk | as a function of ω for i ‰ k will converge in general. However, as described by Ahlfors, it is possible to modify (63) by subtracting an analytic function p n from each singular part P n , where each p n can be chosen as a polynomial. In the case where Ω is the complex plane, then, in Theorem 4 of [2, pg. 187], Ahlfors proves that every meromorphic function has a development in partial fractions and that the singular parts can be described arbitrarily, with this being a particular case of a more general result due to Mittag-Leffler. In particular, he explains that the modified series f pwq " gpzq`ÿ n P nˆ1 w´a n˙´p n pwq, can constructed by taking p n to be the Taylor series expansion of P n´1 w´an¯e xpanded about 0 and truncated at some sufficient degree n v . Still further, he explains that the series in (64) can be made absolutely convergent in the whole complex plane, apart from the poles, by choosing n v sufficiently large, in particular such that 2 nv ě M n 2 n for all n where M n " max |P n pwq| for |w| ă a n {2.
We apply this result to M ij pwq with w :" iν and obtain the following theorem, which is the main result of this section.
Theorem 8.13. The coefficients of Mpwq are meromorphic in the whole complex plane with simple poles at λ n on the positive real axis, where 0 ă λ 1 ă λ 2 ă . . ., and is analytic at w " 0 with Mp0q " N 0 . Thus, the coefficients of Mpwq admit a Mittag-Leffler type expansion for simple poles in the form where p n pwq "´ˆw λ n`w 2 λ 2 n`.
pA pnq q ij :"´α 3 λ n 4 multpλnq ÿ k"1 φ n,k , Θ p0q pe i q In the above, pλ n , φ n q are the eigenvalue-eigensolution pairs of (38). The series can be made absolutely convergent in the complex plane, apart from at the poles, by choosing n v sufficiently large, in particular such that 2 nv ě M n 2 n for all n where M n " max | pλ n {pw´λ n q`1q A pnq ij | for |w| ă λ n {2.
Proof. Recall R σ˚`i I σ˚" M´N 0 and from (51), that for objects B with homogenous σ˚, and possibly inhomogeneous µ˚, where we have introduced w " iν. Thus, by introducing (67), we have We recall from Lemma 4.1 that for the limiting case of ν " 0 we have Mp0q " N 0 , which, by Corollary 4.2, reduces to the Póyla-Szegö tensor when considering a single object with homogeneous µ˚, and as its coefficients are independent of ν they are clearly analytic. Thus, f pwq " pMpwqq ij is of the form of (63) with gpwq " pN 0 q ij and P n pwq "´λ n w´λn`1¯p A pnq q ij where the poles are simple. We already know from Lemma 8.5 that (68) is convergent for ν P r0, 8q, i.e. when w lies on the positive imaginary axis, away from the poles in the real axis. We can extend this further by applying the Mittag-Leffler Theorem, described above, and constructing a modified expansion (65) where our p n pwq in (66) is the Taylor series expansion of our P n pwq about 0 and truncated at n v in such a way to ensure that it is convergent at all points in the complex plane away from the poles. This then immediately leads to our quoted result.
Corollary 8.14. Expanding Mpsq " N 0`Rσ˚p sq`iI σ˚p sq in terms of s "´iω we have that pMpsqq ij is meromorphic in the whole complex plane with simple poles at s n "´λ n {pµ 0 σ˚α 2 q on the negative real axis where 0 ă |s 1 | ă |s 2 | ă . . . and is analytic at s " 0 with Mp0q " N 0 and, hence, in the case of n v " 0, admits the expansion which is absolutely convergent in the whole complex plane, apart from the poles, provided that max | ps n {ps´s n q`1q A pnq q ij |, for |s| ă |s n |{2, decays faster than 2´n.
Proof. For n v " 0 the result stated in (65) in Theorem 8.13 becomes which is convergent for ν P r0, 8q, i.e. when w lies on the positive imaginary axis and is absolutely convergent in the whole complex plane, apart from the poles provided that M n " max | pλ n {pw´λ n q`1q A pnq q ij | for |w| ă λ n {2 decays faster than 2´n . Still further, using a simple change of variables, we can obtain an expansion of Mpsq " N 0`Rσ˚p sq`iI σ˚p sq in terms of s "´iω and find that the poles are at s n "´λ n {pµ 0 σ˚α 2 q on the negative real axis where 0 ă |s 1 | ă |s 2 | ă . . .. Making the change of variables w "´sµ 0 σ˚α 2 in (70) gives (69).
Remark 8.15. Wait and Spies [33] obtained an analytical solution for a conducting permeable sphere and obtained explicit expressions for the tensor coefficients and the negative real values of the poles s n for this case. Their choice of s n corresponds to our s n µ r in the case of a permeable homogeneous object, but ours is more general as it can also be applied to inhomogeneous objects where µ r is no longer a constant. For other shapes with homogeneous parameters, Baum [9] has predicted that Mpsq has simple poles on the negative real axis and quoted Mpsq "Mp0q`8 ÿ n"1 s s n ps´s n q M n M n b M n`p ossible entire function, s´s n`1 s n˙M n M n b M n`p ossible entire function, 9 Transient response of pH α´H 0 qpxq Building on the earlier work of Wait and Spies [33], who have obtained an analytical expression for transient response from a conducting permeable sphere, we can now apply Theorem 8.13 to obtain explicit expressions for the transient response from an inhomogeneous conducting permeable object with σ˚fixed.
Theorem 9.1. The transient perturbed magnetic field response to B α with fixed σ˚placed in a background field H step 0 px, tq " H 0 pxquptq is pH α´H step 0 qpx, tq i "pD 2 Gpx, zqq ij pM step ptqq jk pH 0 pzqq k`p Rpx, tqq i , pM step ptqq jk :"˜pN 0 q jk`8 ÿ n"1 e snt pA pnq q jk¸u ptq, where H 0 pxq is real valued and uptq a unit step function, generated by a divergence free current source of the form J step 0 px, tq " J 0 pxquptq with real valued J 0 pxq. In the above, H α px, tq is the transient magnetic interaction field, which satisfies the transient version of (3), s n " λ n {pµ 0 σ˚α 2 q, λ n is an eigenvalue of (38) and A pnq ij is as defined in (67). If the conditions of the asymptotic formula (4) are met then Rpx, tq " 0.
Proof. For consistency with Wait and Spies [33] we set s " iω and apply where c is a positive constant and L´1 denotes the inverse Laplace transform. The complex conjugate of pH α´H 0 qpxq i is taken as Wait and Spies use e iωt rather than e´i ωt used here. Now, substituting the asymptotic formula (4), we have, assuming H 0 is real, that By considering (65), applying the change of variables from w " iν to s "´iω, so that the poles lie on the negative real axis at s n "´λ n {pµ 0 σ˚α 2 q, as discussed in Corollary 8.14, and closing the contour by an infinite semicircle in the left hand s plane, we find, for t ą 0, that . .`s nv s nv n q and we have used the fact that s n is real. For t ă 0, we close the integral by an infinite semicircle in the righthand s plane and find that the integral vanishes in this case as there are no poles in the right hand plane. From [5], under the conditions of (4) are met, then Rpx, sq ď Cνα 4 }H 0 } W 2,8 pBαq " C|s|µ 0 σ˚α 6 }H 0 } W 2,8 pBαq and, hence, pRpx, tqq i " 1 2πi ż c`i8 c´i8 1 s pRpx, sqq i e st ds " L´1ˆ1 s pRpx, sqq i˙" 0.
Thus, the result immediately follows.
Remark 9.2. Theorem 9.1 shows the long-time response of the perturbed field for a step function characterises of an inhomogeneous object by the N 0 tensor, which describes the magnetostatic characteristics of B α . Similar observations were found for a conducting permeable sphere by Wait and Spies [33]. Despite the issues with the convergence of Baum's [9] for a homogenous object it leads to a predication that is similar to that obtained in (71) when the correct form of Mittag-Leffler theorem is used. However, importantly, all terms are now explicitly defined and, under the conditions of (4), can be computed. where H 0 pxq is real valued and δptq is a delta function associated with an impulse at t " 0, generated by a divergence free current source of the form J imp 0 px, tq " J 0 pxqδptq with real valued J 0 pxq. In the above, H α px, tq is the transient magnetic interaction field, which satisfies the transient version of (3), s n "´λ n {pµ 0 σ˚α 2 q, λ n is an eigenvalue of (38) and pA pnq q ij is as defined in (67). If the conditions of the asymptotic formula (4) are met then Rpx, tq " 0.
Remark 9.4. Theorem 9.3 shows that the short-time response of the perturbed field for an impulse function characterises an inhomogeneous object by the coefficients of the Mp8q tensor, which describes a perfectly conducting object B α . Similar observations were found for a conducting permeable sphere by Wait and Spies [33]. This is also confirms Baum's [9] predication for homogeneous conducting objects and makes explicit all of the terms if the conditions of the asymptotic formula (4) are met.
Remark 9.5. Theorems 9.1 and 9.3 rely on the conditions of the asymptotic formula (4) in order for Rpx, tq to vanish. In general, when these conditions are not met, we do not have an estimate of Rpx, tq. Quantifying its behaviour for more general circumstances will form part of our future work.