Bruno Després

Lise-Marie Imbert-Gérard

Olivier Lafitte

Singular solutions for the plasma at the resonance *

Little is known on the mathematical theory of hybrid and cyclotron solutions of the Maxwell equations with the cold plasma dielectric tensor. Such equations arise in magnetic plasmas such the ones needed for the modeling the an electromagnetic wave in Tokamaks. The behavior of solutions can be extremely different to those in vacuum. This work intends to contribute to the local theory by means of original representation formulas based on special functions and a certain eikonal equation, and with a careful treatment of the singularity for the hybrid resonance.

Introduction

The goal of this work is to study with an original mathematical method some singular solutions of the cold plasma model at different resonances, singular meaning here that at a point x * a function u satisfies |(x -x * )u(x)| ≥ c for some c > 0. In a more general sense, the function u can also be a Dirac mass or a principal value. The problem comes from the modeling of electromagnetic waves in strongly magnetized plasmas and from the socalled resonant heating phenomenon. We more specifically consider the one species cold plasma model which represents a collection of zero-temperature electrons immersed in a uniform static magnetic field. Resonances correspond to limits of this model, and will be studied in this work thanks to the introduction of the friction of electrons on the background ions. In slab geometry in dimension two, (x, y) ∈ R 2 , and time harmonic regime, there is no resonance in the equation for the transverse magnetic polarization (also denoted as O-mode equations in plasma physics), while the equations for the transverse electric polarization (also denoted as X-mode) have two resonances. This work focuses on the latter equations, that can be reduced to

-∂ 2 y E 1 +∂ 2 xy E 2 -ǫ 11 E 1 -ǫ 12 E 2 = 0, ∂ 2 xy E 1 -∂ 2 x E 2 -ǫ 21 E 1 -ǫ 22 E 2 = 0, (1)
where the electric field (E 1 , E 2 ) is transverse to the bulk magnetic field. Denote by ω the pulsation of the wave.

A first order approximation as in [START_REF] Stix | The Theory of Plasma Waves[END_REF][START_REF] Swanson | Plasma Waves[END_REF][START_REF] Chen | Amplification and Absorption of Electromagnetic Waves in Overdense Plasmas[END_REF][START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF] through the cold plasma approximation yields the following dielectric tensor

ǫ =   1 - ω 2 p (ω 2 -ω 2 c ) i ωcω 2 p ω(ω 2 -ω 2 c ) -i ωcω 2 p ω(ω 2 -ω 2 c ) 1 - ω 2 p (ω 2 -ω 2 c )   , (2)
where the cyclotron frequency is defined by ω c = e|B0| me while the plasma frequency plasma frequency is defined by ω p = e 2 Ne ǫ0me . Assume that the magnetic field and the electronic density depend on the horizontal variable x only: that is B 0 = B 0 (x) and N e = N e (x). It corresponds to a plasma facing a wall as depicted in figure 1. This is typically what occurs at a wall in a Tokamak where an antenna sends an electromagnetic wave toward the plasma in order to probe or to heat the plasma. Heating of a magnetic fusion plasma with such devices x Antenna x=-L reflected wave incident wave Figure 1: X-mode in slab geometry: the domain. In a real physical device an antenna is on the wall on the left and sends an incident electromagnetic wave through a medium which is assumed to be infinite for simplicity. The incident wave generates a reflected wave which can be used to identify the properties of the plasma. The medium is filled with a plasma with dielectric tensor given by [START_REF] Branbilla | Kinetic Theory of Plasma Waves-Homogeneous Plasmas[END_REF]. is important issue for the ITER project [START_REF]ITER organization[END_REF]. The physical coefficients at the wall of a Tokamak are described in figure 2. More realistic plasma parameters may be much more oscillating as shown in the numerical studies [START_REF] Da | Stable coupling of the Yee scheme with a linear current model[END_REF][START_REF] Silva | A Numerical Study of Forward-and Backscattering Signatures on Doppler-Reflectometry Signals[END_REF].

x B 0 x=-L N e

Figure 2: The physical parameters for the X-mode equations in slab geometry. The electronic density x → N e (x) is low at the boundary, and increases towards a plateau. The background magnetic field B 0 is here taken as constant for simplicity.

Resonances correspond to a limit of the geometric optics, as the local wave number approaches infinity. They require a careful examination since they are related to singular behavior and absorption. The interested reader can refer to the physics textbooks [START_REF] Stix | The Theory of Plasma Waves[END_REF][START_REF] Swanson | Plasma Waves[END_REF], or to [START_REF] Chen | Amplification and Absorption of Electromagnetic Waves in Overdense Plasmas[END_REF] for a historical physical paper on the topic. A recent physical reference is [START_REF] Dumont | Effects of non-Maxwellian species on ion cyclotron waves propagation and absorption in magnetically confined plasmas[END_REF] and therein. See also for singular solutions of wave equations for the theory of cloaking [START_REF] Weder | A Rigorous Analysis of High-Order Electromagnetic Invisibility Cloaks[END_REF] and for metamaterials [START_REF] Chen | Resonance and Double Negative Behavior in Metamaterials[END_REF][START_REF] Bonnet-Ben Dhia | T -coercivity for scalar interface problems between dielectrics and metamaterials[END_REF]. As announced earlier, it is then natural to model the effect of collisions with a bath of static ions by an additional friction term, since the energy is absorbed by the ions. Describing the behavior of the solutions as the friction parameter ν > 0 approaches zero will evidence the singularity of the solutions to the limit problem for ν = 0. The singular solutions will be built by means of new explicit representation formulas which allow a careful extraction of the singularities. The singular behavior of the limit solutions is strongly related to the structure of the regularized dielectric tensor. The complex pulsation ω = ω + iν shifts the exact pulsation in the upper half-plane by the factor ν > 0.

where ω c is a function of the x variable, a priori non constant. We will show that mathematical solutions of Maxwell's equations [START_REF] Budden | Radio Waves in the Ionosphere[END_REF] actually have no singular behavior: they are bounded in this regime, despite the fact that the dielectric coefficients are not. The other interesting regime corresponds to the vanishing of the diagonal part of ǫ 0 , that is to say ω 2 p (x) + ω 2 c (x) = ω 2 . This is the hybrid resonance. In this regime the dielectric tensor is non singular since the denominator is non zero, i.e. the coefficients of the dielectric tensor are bounded. However the solutions are known to be highly singular: they are not bounded near the resonance. As stressed in [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF], a useful quantity that can be used to characterize the singular behavior of the solutions is the resonant heating of the plasma. The resonant heating [START_REF] Branbilla | Kinetic Theory of Plasma Waves-Homogeneous Plasmas[END_REF][START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF] is defined by

Q = lim ν→0 + ℑ (ǫ ν E ν • E ν ) dx (7)
where the total electric field is E ν = (E ν 1 , E ν 2 ) and (A • B) = A•B denotes the hermitian product of the complex vectors A and B. This formula has been justified in [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF] where it is shown that Q > 0 in case of resonant heating using the simplification ǫ ν = ǫ + iνI. It means that the dissipation tensor introduced in the ν expansion of ǫ ν (5) is diagonal and D = I. This simplification is mathematically justified since it corresponds to the limiting absorption principle. In this work we will show that the resonant heating is well defined and takes the same value for a large class of dissipation tensors that includes physically based tensors. The method of analysis uses

W ν = ∂ x E ν 2 -∂ y E ν
1 which is, after linearization around the bulk magnetic field, the parallel component of the magnetic field. The system (4) is then equivalent to the following first order system

   ∂ y W ν -ǫ ν 11 E ν 1 -ǫ ν 12 E ν 2 = 0, -∂ x W ν -ǫ ν 21 E ν 1 -ǫ ν 22 E ν 2 = 0, W ν +∂ y E ν 1 -∂ x E ν 2 = 0.
Thanks to the fact that the coefficients ǫ ν ij depend only on x, using the Fourier transform in y, denoting by iθ the Fourier coefficient, it yields

   iθW ν -ǫ ν 11 E ν 1 -ǫ ν 12 E ν 2 = 0, -d dx W ν -ǫ ν 21 E ν 1 -ǫ ν 22 E ν 2 = 0, W ν +iθE ν 1 -d dx E ν 2 = 0. (8)
The Fourier coefficient θ can be considered as a frozen parameter from now on, and ∂ x will be replaced by d dx for the derivative with respect to x. In this system, the derivative with respect to x appears only on W ν and E ν 2 . The energy integral [START_REF] Coddington | Theory of ordinary differential equations[END_REF] can be expressed differently using the identity b a

(ǫ ν E ν • E ν ) dx = b a iθW ν E ν 1 - d dx W ν E ν 2 dx = b a iθW ν E ν 1 + W ν d dx E ν 2 dx -W ν (b)E ν 2 (b) + W ν (a)E ν 2 (a) = - b a |W ν | 2 dx -W ν (b)E ν 2 (b) + W ν (a)E ν 2 (a).
One obtains another expression

ℑ b a (ǫ ν E ν • E ν ) dx = -ℑ W ν (b)E ν 2 (b) + ℑ W ν (a)E ν 2 (a) (9)
which expresses a balance of energy.

It is convenient to consider a differential system with only two unknowns, namely (E ν 2 , W ν ), while E ν 1 can be computed thanks to the first equation of ( 8) by

E ν 1 = iθW ν -ǫ ν 12 E ν 2 ǫ ν 11 (10)
The reduced system then reads

d dx E ν 2 (x) W ν (x) = M ν (x, θ) E ν 2 (x) W ν (x) , x ∈ R, (11)
where the matrix is defined as follows.

Definition 1. The matrix M ν (x, θ) is constructed from the dielectric tensor by

M ν (•, θ) =      - iθǫ ν 12 ǫ ν 11 1 - θ 2 ǫ ν 11 - d ν ǫ ν 11 - iθǫ ν 21 ǫ ν 11      (12)
where the coefficient d ν is the determinant of the dielectric tensor ǫ ν

d ν = ǫ ν 11 ǫ ν 22 -ǫ ν 12 ǫ ν 21 . ( 13
)
The determinant of M ν (x, θ) will be denoted

D ν (•, θ) = det M ν (•, θ).
The determinant of the matrix can be simplified as follows

D ν (•, θ) = - iθǫ ν 21 ǫ ν 11 2 + d ν ǫ ν 11 1 - θ 2 ǫ ν 11 = d ν ǫ ν 11 -θ 2 ǫ ν 22 ǫ ν 11 = d ν ǫ ν 11 -θ 2 . ( 14
)
It is already foreseeable that the roots of ǫ ν 11 will play a crucial part in some cases: the analysis of singular solutions relies on the analyticity of the coefficients of the dielectric tensor, at least locally around x = x * , where x * is a hybrid resonance. This assumption is very useful to study the problem with a convenient shift in the complex plane. The main results of this paper stem from a precise local study of the solution of the reduced system when the entries of M ν have a simple pole at x * , and the determinant of the matrix has a simple pole as well. The mathematical method of singularity extraction is completely new to our knowledge. Our main results can be summarized as follows:

Lemma 2. In our geometry, the cyclotron singularity of the dielectric tensor (ω = ω c ) is only an apparent singularity in the sense that the electromagnetic field is bounded and the limit heating is zero Q cyc = 0.

Theorems 1 and 2. The hybrid resonance (ω 2 = ω 2 c + ω 2 p ) yields a singular horizontal component of the electric field, i.e. |(x -x * )E ν=0 1 | ≥ c > 0, and a generic positive resonant heating Q hyb > 0. The limit of the horizontal component of the electric field can be expressed as the sum of a Dirac mass, of a Principal Value and of a bounded term, as in [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF].

Corollary 1. The value of the hybrid resonant heating is independent of the local dissipation tensor, and moreover the local dissipation tensor can be non physical as well. The limit value for ν = 0 + of the electromagnetic unknowns is independent of the local dissipation tensor. It establishes the uniqueness of the limit solution with respect to the local dissipation tensor. This is a new result that was not possible with the techniques developed in [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF] where the limit was established only for D = I.

Numerical section

We provide numerical evidence of our claims with the help of a numerical method that can be used to treat other problems with same structure.

Extension to multi-species We finally provide some formulas which shows the method of singularity extraction can be used for multi-species, such as ions and electrons. These formulae are easier to handle than the sum of dielectric tensors generally used both for the cyclotron resonances and the hybrid resonances.

The work is organized as follows. The cyclotron singularity of the dielectric tensor is studied in section 2, where the electromagnetic field is shown to stay bounded in the neighborhood of this singularity. The hybrid resonance is studied in detail in section 3 where the singularity is carefully extracted using convenient representation formulas with Bessel functions. We prove both the generic resonant heating and the fact that its value is independent of the local dissipation tensor. Numerical results are provided in section 4. Some alternative formulas are collected in the appendix.

The cyclotron resonance

The cyclotron singularity corresponds to a formal singularity in the denominator of (3). We consider the situation where there exists at least one x c in the domain such that ω c (x c ) = ω. For the simplicity of the analysis, we will assume that d dx ω c (x c ) = 0 so that x c is isolated. Clearly the dielectric tensor (3) is singular at x c at the limit ν = 0 + , while it is not singular for the regularized system, i.e. ν = 0. In order to focus on the cyclotron resonance, we want it to be away from the hybrid resonance, that is ω p = 0. This is a generic case for a plasma. Definition 2. A point x c is referred to as an isolated cyclotron singularity if there exists r > 0 such that ω c (x c ) = ω and d dx ω c (x c ) = 0. Note that such a singularity does not depend on ν, since ω c does not depend on ν. Let us now consider the reduced system [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF], where the system's matrix M ν (•, θ) is defined with the physical dielectric tensor (3). We assume there is an isolated singularity x c . Then the system is well defined for all ν ∈ R, and one can pass to the limit in the following sense.

Lemma 1. Assume that the magnetic field B 0 and the density N e are smooth. Assume an isolated cyclotron singularity at x c , and assume there is no hybrid resonance on [x c -r, x c + r]. Assume (E ν 2 , W ν ) solves the well posed system (11) on [x c -r, x c + r] for all ν ∈ R, with a given Cauchy data which admits a finite limit when ν goes to 0 written as

(E ν 2 , W ν )(x c -r) = (a, b). Then (E ν
2 , W ν ) tends to a bounded limit (E 2 , W ) as ν tends to zero, and (E 2 , W ) solves the limit system (11) on [x c -r, x c + r] with ν = 0 and with the same Cauchy data (E 2 , W )(x c -r) = (a, b)

Proof. Definition 1 with the physical tensor (3) yields after some careful calculations

M ν (•, θ) = 1 α ν (•, θ)    θω c ω 2 p (1 -θ 2 )ω(ω 2 -ω 2 c ) -ωω 2 p -ω(ω 2 -ω 2 c ) + 2ωω 2 p - ω 4 p ω -θω c ω 2 p    (15)
where

α ν (•, θ) = ω(ω 2 -ω 2 c ) -ωω 2 p .
The determinant [START_REF] Imbert-Gérard | Analyse mathématique et numérique de problémes d'ondes apparaissant dans les plasmas magnétiques[END_REF] of this matrix can be recast as

D ν (•, θ) = d ν ǫ ν 11 -θ 2 = ω(ω 2 -ω 2 c )-2ωω 2 p + ω 4 p ω αν (•,θ) -θ 2 .
The limit value of α as ν approaches 0

+ is α 0 (•, θ) = ω ω 2 -ω 2 c -ω 2 p .
Since the cyclotron resonance is not also an hybrid resonance, ω p > 0, one has that

α 0 (x c , θ) = -ωω 2 p = 0 since ω = ω c (x c ) = 0. ( 16
)
In this situation the matrix M ν (•, θ) is locally bounded and smooth. So the solutions of the reduced system [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF] are smooth locally around x c and one can pass to the limit in this system.

Note that there is still a particular behavior in the neighborhood of this cyclotron resonance, known as 'turning point behavior' [START_REF] Coddington | Theory of ordinary differential equations[END_REF][START_REF] Lafitte | High-frequency stability of multidimensional ZND detonations[END_REF], but which does not induce any singular regime, except in the limit ω → +∞, which is not considered here. The consequences of the representation (15) on physical quantities are characterized in the following result.

Lemma 2. Under the assumptions of Lemma 1, then E ν 1 also tends to a bounded limit E 1 and the value of the limit heating is zero

lim ν→0 + Im xc+r xc-r (ǫ ν E ν • E ν ) dx = 0. ( 17
)
Proof. The horizontal component of the electric field can be evaluated using [START_REF] Dumont | Effects of non-Maxwellian species on ion cyclotron waves propagation and absorption in magnetically confined plasmas[END_REF]: after some computations the coefficients in [START_REF] Dumont | Effects of non-Maxwellian species on ion cyclotron waves propagation and absorption in magnetically confined plasmas[END_REF] can be expressed as

iθ ǫ ν 11 = iθω ω2 -ω c (•) 2 α ν (•, θ) and -ǫ ν 12 ǫ ν 11 = -iω c (•)ω 2 p α ν (•, θ) .
Using ( 16), the limit ν = 0 value of these coefficients at x c is

iθ ǫ 0 11 (x c ) = 0 and -ǫ 0 12 ǫ 0 11 (x c ) = i. (18)
So one can pass to the limit in [START_REF] Dumont | Effects of non-Maxwellian species on ion cyclotron waves propagation and absorption in magnetically confined plasmas[END_REF] since the coefficients are bounded in the interval [x c -r, x c + r]. Therefore the horizontal part of the electric field E ν 1 admits a bounded and smooth limit E 1 . Consider now the heating term. On the first hand we notice that the electric field can be split into two parts

E ν = E ν 2 i 1 + E ν 1 -iE ν 2 0 ,
the first vector being an eigenvector of the dielectric tensor :

ǫ ν i 1 = λ i 1
where the eigenvalue

λ = 1 - ω 2
p ω(ω+ωc) is bounded so non singular at x c . The second vector is O (ν + |x -x c |) due to [START_REF] Dumont | Effects of non-Maxwellian species on ion cyclotron waves propagation and absorption in magnetically confined plasmas[END_REF] and [START_REF] Swanson | Plasma Waves[END_REF], which means that it counterbalances the singularity of the dielectric tensor. It shows that ǫ ν E ν is bounded uniformly on [x c -r, x c + r] with respect to ν. Since E ν is also uniformly bounded, it yields that (ǫ ν E ν , E ν ) is bounded uniformly on [x c -r, x c + r] with respect to ν. On the other hand the tensor ǫ ν tends almost everywhere (that is for x = x c ) to an hermitian tensor, so that ℑ (ǫ ν E ν , E ν ) tends almost everywhere to zero. The Lebesgue dominated convergence theorem states that lim

ν→0 + ℑ xc+r xc-r (ǫ ν E ν • E ν ) dx = 0.
The proof is ended.

It evidences the fact that there is no resonant cyclotron heating in our model. This result is compatible with the literature [START_REF] Stix | The Theory of Plasma Waves[END_REF][START_REF] Swanson | Plasma Waves[END_REF][START_REF] Chen | Amplification and Absorption of Electromagnetic Waves in Overdense Plasmas[END_REF] where a resonant cyclotron heating is possible, but only in another configuration with wave number parallel to the bulk magnetic field. In our case it yields ∂ x = 0 and so is excluded from our discussion.

The hybrid resonance

This section gathers the main theoretical contributions of this work. The required hypothesis on the general dielectric tensor

ǫ ν (x, y) = ǫ ν 11 (x) ǫ ν 12 (x) ǫ ν 21 (x) ǫ ν 22 (x) ∈ C 2×2 , ∀ν ∈ R, (19)
for the following work to hold will be emphasized in Assumptions 1, 2 and 3. Note that each one of these assumptions is satisfied by the physical tensor ( 3), but that all the following results hold for the wider range of tensors satisfying these three assumptions. The hybrid resonance corresponds to a singularity in the limit problem ν = 0, it more precisely corresponds to the vanishing of ǫ 0 11 , which occurs in the expression of the x component of the electric field E 0 1 , see [START_REF] Dumont | Effects of non-Maxwellian species on ion cyclotron waves propagation and absorption in magnetically confined plasmas[END_REF], as well as in the matrix of the (E 0 2 , W 0 ) system, see [START_REF]ITER organization[END_REF] with ν = 0.

Definition 3. A point x h is referred to as a local hybrid resonance for the limit system (4)-( 19), i.e. with ν = 0, if the limit diagonal coefficient of ǫ ν 11 (x), is such that

ǫ 0 11 (x h ) = 0 and ∂ x ǫ 0 11 (x h ) ∈ R * (20)
and if the extra-diagonal part is locally non zero

ǫ 0 12 (x h ) = 0. ( 21
)
We will moreover assume that

i∂ ν ǫ 0 11 (x h ) ∈ R * (22)
which is compatible with the physical context as visible in equation ( 6).

Going back to the physical dielectric tensor (3), one sees that ǫ 0 11 (x h ) = 0 corresponds to

ω 2 = ω 2 c (x h ) + ω 2 p (x h ).
This case is referred to as hybrid regime in the literature, see [START_REF] Stix | The Theory of Plasma Waves[END_REF][START_REF] Swanson | Plasma Waves[END_REF][START_REF] Chen | Amplification and Absorption of Electromagnetic Waves in Overdense Plasmas[END_REF]. Unlike in the cyclotron resonance case, there exist singular solutions at the hybrid resonance. This is known since the seminal work of Budden [START_REF] Budden | Radio Waves in the Ionosphere[END_REF], for an extremely particular case where ǫ 0 11 = 1 + x x h . A recent mathematical analysis, performed in [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF] with a singular integral equation, has evidenced the role of the limit resonant heating. Our main objective of the present work is to provide additional understanding of the resonant heating by means of a local extraction of the singularity. Stemming directly from the definition, a first property of the 2 × 2 differential system [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF] is the following: Lemma 3. Assume x h is local hybrid resonance, then for ν = 0 the determinant (13) of the dielectric tensor is such

d 0 (x h ) = -ǫ 0 12 (x h ) 2 < 0 (23)
Hereafter follows the list of some assumptions on the general regularized tensor [START_REF] Tricomi | Integral equations[END_REF]. This is required for the upcoming analysis.

Assumption 1 (Uniqueness and Analyticity of the coefficients). Assume x h is a local hybrid resonance. There exists Λ 1 > 0 such that

• the dielectric tensor ǫ ν is analytic in the ball x ∈ B(x h , Λ 1 ) for all ν ∈ [-Λ 1 , Λ 1 ],
• x h is the unique root of ǫ 0 11 in B(x h , Λ 1 ). Note that, for F satisfying this assumption of analyticity,

F (z) = n≥0 a n z n , with |a n |R n < +∞ for R < Λ 1 .
One shall denote the derivative as usual by F ′ (z) = n≥0 na n z n-1 . One has the chain rule d dx (F (φ(x))) = F ′ (φ(x)) dφ dx (x) for φ a function from an interval I of R to C. Assumption 2 (Antisymmetry of the dielectric tensor). The dielectric tensor ǫ ν is such that

ǫ ν 12 = -ǫ ν 21 , ∀ν ∈ [-Λ 1 , Λ 1 ]. ( 24
)
As a result, the trace of the matrix M ν (x, θ) of the 2 × 2 system is identically zero on its domain.

The last assumption is satisfied by the physical tensor (2) and by its first order approximation [START_REF] Chen | Resonance and Double Negative Behavior in Metamaterials[END_REF]. Also note that as a consequence of this assumption, the determinant of M ν is less singular than expected: since all the entries of the matrix are O((ǫ ν 11 ) -1 ), the determinant would be expected to be O((ǫ ν 11 ) -2 ), but from (24) it actually is only O((ǫ ν 11 ) -1 ). For the sake of clarity, consider the following notation:

Definition 4. Define for all ν ∈ [-Λ 1 , Λ 1 ] a ν = - iθǫ ν 12 ǫ ν 11 , b ν = 1 - θ 2 ǫ ν 11 , c ν = - d ν ǫ ν 11 , (25)
so that thanks to the symmetry property (24), the matrix M ν introduced in Definition 1 reads

M ν (x, θ) = a ν (x, θ) b ν (x, θ) c ν (x) -a ν (x, θ) . ( 26
)
These quantities are well defined except possibly at the roots of ǫ ν 11 . In the model case ǫ ν = ǫ 0 + iνD with a diagonal dissipation tensor D = I studied in [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF], one has ǫ ν 11 = α(x) + iν and ǫ ν 12 = iδ(x). In this case the coefficients are exactly

a ν (x, θ) = θδ(x) α(x)+iν , b ν (x, θ) = 1 - θ 2 α(x)+iν and c ν (x) = δ 2 (x)
α(x)+iν -(α(x) + iν), and so c ν does not depend on θ.

Assumption 3. The regularized dielectric tensor coefficient ǫ ν 11 (z) is locally analytic with respect to both variables z and ν.

The techniques developed in this work are direct consequences of these properties. The study that follows is performed in two steps: the regularized problem, with ν = 0 is first analyzed, then the limit as ν approaches zero is studied, to go back to the original problem. To this purpose it is crucial to develop uniform tools with respect to the regularization parameter ν, so that their properties still hold as ν goes to zero.

A convenient second order equation

System [START_REF]ITER organization[END_REF] with matrix (26) can provide two different second order equations, by elimination of either of the two unknowns. In order to choose which of these two equations to study, consider the following consequence of Definition 3.

Lemma 4. Assume that x h is a local hybrid resonance. Then the dielectric tensor ǫ ν (3) is such that:

• There exists Λ 2 satisfying 0 < Λ 2 ≤ Λ 1 and C > 0 independent of the angle of incidence θ such that

a ν (x) c ν (x) ≤ C|θ| for x ∈ B(x h , Λ 2 ) and for |ν| ≤ Λ 2 . (27)
• The quantity aν

(x h ) bν (x h ) = ǫ ν 12 (x h ) θ is unbounded for θ → 0.
The fact that the upper extra diagonal coefficient b ν does not dominate the diagonal part a ν for vanishing θ results in a singularity in this regime which makes the analysis trickier than with for the equation on E ν 2 , therefore the formulation considered is the one on W ν :

d dx 1 c ν d dx W ν = a 2 ν c ν + b ν - a ν c ν ′ W ν . ( 28
)
Remark 1. At the resonance, the limit equation (28) has a 'regular singular point' [START_REF] Coddington | Theory of ordinary differential equations[END_REF][START_REF] Lafitte | High-frequency stability of multidimensional ZND detonations[END_REF]. Indeed, thanks to Definition 3 and Lemma 4, the 1 cν term is bounded while the An adequate scaling of the unknown W ν then provides an equation with no first order term. The rescaled unknown y

(x) = 1 √ -cν (x)
W ν (x) indeed satisfies the equation

d 2 y dx 2 = a 2 ν + b ν c ν -c ν a ν c ν ′ + √ -c ν 1 √ -c ν ′′ y, x ∈ R. ( 29
)
Remark 2. In this whole work, the square root has to be understood as the principal square root on complex numbers.

A consequence of Assumption 3 provides a unique complex root of the coefficient ǫ ν 11 in a neighborhood of the root of the limit coefficient ǫ 0 11 , thanks to the open mapping theorem: Lemma 5. Suppose x h is a local hybrid resonance. Under Assumptions 1, 2 and 3, there exists Λ 3 satisfying

0 < Λ 3 ≤ Λ 2 such that if |ν| ≤ Λ 3 there exists a unique point X ν ∈ B(x h , Λ 3 ) ⊂ C in a neighborhood of x h solution of ǫ ν 11 (X ν ) = 0. ( 30
)
A first order expansion is

X ν = x h - ∂ ν ǫ 0 11 (x h ) ∂ x ǫ 0 11 (x h ) ν + O(ν 2 ) (31)
which shows the limit of X ν as ν approaches zero is x h and X ν is non real for small ν = 0 in view of (22).

Proof. From Theorem 10.30 of [START_REF] Rudin | Real and complex analysis[END_REF] and Definition 3, there exists neighborhoods V 0 and V ν of x h on which respectively ǫ 0 11 and ǫ ν 11 -ǫ ν 11 -ǫ 0 11 (x h ) are one-to-one. Assumption 3 then implies that there exists a Λ 4 > 0 such that for all ν satisfying |ν| < Λ 4 , -ǫ ν 11 -ǫ 0 11 (x h ) ∈ V ν , and that there exists Λ 5 > 0 such that for all

ν satisfying |ν| < Λ 5 , (ǫ ν 11 ) ′ (x h ) = 0. Set Λ 3 = min(Λ 2 , Λ 4 , Λ 5 ).
Then for all ν satisfying |ν| < Λ 3 there is a unique solution of the equation

ǫ ν 11 (x) -ǫ ν 11 -ǫ 0 11 (x h ) = -ǫ ν 11 -ǫ 0 11 (x h ), ∀x ∈ V ν , that is to say a unique X ν ∈ V ν satisfying Equation (30). The formula (31) is a consequence of the local expansion 0 = ǫ ν 11 (X ν ) = ∂ x ǫ 0 11 (x h )(X ν -x h ) + ∂ x ǫ 0 11 (x h )ν + O(ν 2 ).
Definition 5. The unique point X ν ∈ C solution of (30) is called the translated hybrid resonance.

The case ǫ ν 11 = α(x) + iν mentioned above satisfies these hypotheses, with α(0) = 0, α ′ (0) < 0, and for this case one has, at first order, X ν = -i ν α ′ (0) + O(ν 2 ), which means that the dominant part of the translated resonance is pure imaginary with positive imaginary part. In order to solve Equation (29), it is then crucial to isolate the singularity of the solution. We propose here to first understand the singular structure of the equation's coefficient. In the case of a local hybrid resonance as in Definition 3, we notice that the most singular term of the coefficient is

√ -c ν 1 √ -c ν ′′ .
We then consider the following quantity: Definition 6. Suppose x h is a local hybrid resonance. Under Assumptions 1, 2 and 3, we define the 'coefficient function'

R ν (•) in B(x h , Λ 3 ) as: R ν (.) = (• -X ν ) a 2 ν + b ν c ν -c ν a ν c ν ′ + √ -c ν 1 √ -c ν ′′ + 1 4(• -X ν ) 2 .
With this notation, equation (29) reads

d 2 y(x) dx 2 = - 1 4(x -X ν ) 2 + R ν (x) x -X ν y(x), x ∈ R. ( 32
)
Since the function R ν can also be defined by continuity at X µ , as stated in the following result, this form of the equation evidences the singularity of the coefficient. Note that the method described below extends in a more general case than the one of Definition 3, namely when ǫ 0 11

(x h ) = ∂ x ǫ 0 11 (x h ) = 0 and ∂ 2 x 2 ǫ 0 11 (x h ) ∈ R * . In this case one considers k ν the limit, for x → X ν , of (x -X ν ) 2 a 2 ν + b ν c ν -c ν a ν c ν ′ + √ -c ν 1 √ -c ν ′′
which is finite. The value of k ν will generate the family of approximate solutions needed for the subsequent study. Returning to the situation of Definition 3, one has Lemma 6. Suppose x h is a local hybrid resonance. Under Assumptions 1, 4 and 3, there exists Λ 6 satisfying 0

< Λ 6 ≤ Λ 3 such that the coefficient function z → R ν (z) is analytic on a ball z ∈ B(x h , Λ 6 ) uniformly for |ν| < Λ 6 . Moreover R ν (X ν ) = iθ(ǫ ν 12 ) ′ -(ǫ ν 12 ) 2 (ǫ ν 11 ) ′ - (ǫ ν 12 ) ′ ǫ ν 12 + 1 4 (ǫ ν 11 ) ′′ (ǫ ν 11 ) ′ (X ν )
where X ν is the translated resonance. This makes sense by continuity for ν small since (ǫ ν 11 ) ′ (X ν ) = ∂ x ǫ ν 11 (X ν ) = 0 using [START_REF] Weder | A Rigorous Analysis of High-Order Electromagnetic Invisibility Cloaks[END_REF].

The proof of this Lemma is given in Section A. As announced, since R ν (X ν ) is bounded, the singularity is explicit in Equation (32): there is a (x -X ν ) -2 singularity, and only if R ν (X ν ) = 0 then there is also a (x -X ν ) -1 singularity. Since R ν is bounded and analytic, one can formally shift the equation and the unknown in the complex plane. That is we formally consider the function

y(•) = 1 -c ν (• + X ν ) W ν (• + X ν )
and write the equation for this new function. It can be justified that W ν solution of (28) admits a convenient extension in a complex neighborhood of the hybrid resonance x h . It provides a new form of equation ( 29). However this is only formal at this level.

Definition 7 (Shifted equation). The shifted equation writes

d 2 y(z) dz 2 = - 1 4z 2 + R ν (z + X ν ) z y(z), z ∈ C * . (33)
Our strategy if first to construct an explicit quasi-solution of the shifted equation, and second to shift back all quantities. It will construct an explicit quasi-solution of the original equation (32). The modifications to obtained exact solutions of the original equation (32) will be performed in a third step.

Freezing and defreezing the coefficient-function

This section focuses on equation (33) for all ν = 0. It is important to see that the only hypothesis that is required to that purpose is the fact that the function x → R ν (x + X ν ) is bounded. As a consequence we choose to focus on the more general equation

d 2 y(z) dz 2 = - 1 4z 2 + R(z) z y(z), z ∈ C * , (34)
with the only assumption that the function R is continuous in a neighborhood of the origin. We will try to distinguish between the results that only depend on this feature of R and those that are specific to the case

R = R ν (• + X ν ).
It turns out that equation (34) is related to the Bessel equation: to see that, it is sufficient to replace the coefficient function R by its value at the singularity. Indeed, consider the resulting new equation

d 2 y(z) dz 2 = - 1 4z 2 + R(0) z y(z), z ∈ C * . ( 35
)
This equation is presented in the chapter on Bessel functions in the classical textbook [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], see equation (9.1.50).

According to this reference, solutions are then expressed in terms of the Bessel functions J 0 and Y 0 . Even if the equation is singular at the origin, its solutions are locally bounded as stressed below.

Lemma 7. If R(0) = 0, a pair of independent solutions of (35) can be expressed with Bessel functions under the form

z → √ zJ 0 (λ √ z) , z ∈ C, z → √ zY 0 (λ √ z) , z ∈ C,
where λ = 2 -R(0). If R(0) = 0, a pair of independent solutions of (35) can be expressed with the simpler expression

z → √ z z → √ z log (x) .
Remark 3. The function J 0 (z) admits a infinite converging expansion in powers of z 2

J 0 (z) = ∞ k=0 (-1 4 z 2 ) k (k!) 2 , ( 36
)
and the function Y 0 (z) -2 π log z 2 J 0 (z) admits a infinite converging expansion in powers of z 2

T 0 (z) := Y 0 (z) - 2 π log z 2 J 0 (z) = - 2 π ∞ k=0 ψ(k + 1) (k!) 2 (- z 2 4 ) k , ( 37
)
where ψ(k) = k l=1 1/l for all k ∈ N * . Therefore a more convenient pair of independent solutions of (35) also valid for λ = 0 reads

   z → √ zJ 0 (λ √ z) , z ∈ C, z → √ z Y 0 λ √ z - 2 π log λ 2 J 0 λ √ z , z ∈ C. ( 38
)
The reason for using the pair (38) of independent solutions instead of the one proposed in Lemma 7 is that it is uniform with respect to λ, so when applied to the case

R = R ν (• + X ν ) (39)
it will be uniform with respect to ν. Indeed in this case, the parameter λ will depend on ν since then λ = λ ν = 2 -R ν (X ν ). So the second function from Lemma 7 has a logarithmic divergence when λ ν √ z ≈ 0, which means that it diverges if either λ ν ≈ 0 or √ z ≈ 0. On the other hand the logarithmic divergence of the second function of (38) is by construction only for √ z ≈ 0. As a result the functions (38) are well defined for z = 0 as ν goes to zero, as well as for ν = 0. The Bessel function (37) recasts as Y 0 (z) = T 0 (z) + 2 π log z 2 J 0 (z) and the second function in (38) is also

z → √ z T 0 λ √ z + 1 π log (z) J 0 λ √ z . (40)
The Bessel based functions, solutions of the equation with frozen coefficient (35), provide the formal dominant behavior of the solutions of equation (34). Nevertheless the rigorous justification of this statement is not evident, mainly because the frozen coefficient R(0) is only a first order local approximation of the coefficient function R. Indeed the singularity of the 1 4z 2 term is of second order, so any naive iteration technique that would aim at controlling the error of approximation between both may diverge as well like O z -1 . Overcoming this difficulty is the purpose of the next paragraph, which presents a technical process to control these errors. The next idea is to explicit the link between Equation (34) and the frozen coefficient equation ( 35). Yet the singularity of the equation requires specific attention, and following process is two folded: an intermediate equation is introduced to focus in a first step on the singularity, before finally going back to the desired equation, namely (34).

Eikonal equation and stretching function

Consider first another intermediate equation by reintroducing a varying coefficient in Equation (35):

Definition 8 (Streching procedure). Let ρ be an arbitrary smooth function such that ρ(0) = 0 and ρ ′ (0) = 0. We will refer to it as the stretching function. This name comes, in particular, from Mc Kelvey [START_REF] Kelvey | The solutions of second order linear ordinary differential equations about a turning point of order two Trans[END_REF]. Let u be a given function. The stretching of u is the function ũ defined by

ũ(z) = (ρ ′ (z)) -1 2 u(ρ(z)).
The equation for the stretched function is the following.

Lemma 8. Assume ρ is a stretching function and that u solves the frozen coefficient equation (35

). Set s = (ρ ′ ) -1 2 ′′ (ρ ′ ) -1 2
. Then the stretched unknown y = ũ satisfies

d 2 y(z) dz 2 = - ρ ′ (z) 2 4ρ(z) 2 + R(0)ρ ′ (z) 2 ρ(z) + s(z) y(z), z ∈ C * . ( 41
)
Proof. The proof relies on the scaling of the stretched unknown: the coefficient of the first order derivative term in the computation of ũ′′ is ((ρ ′ ) -1/2 ) ′ ρ ′ + ((ρ ′ ) 1/2 ) ′ , which happens to be zero.

Note that the function s is a priori bounded around 0 since ρ ′ (0) = 0, so that this form of equation evidences again, as Equation (34), the singularities of its coefficient. At this point the stretching function is a tool that will be fit toward the approximation of Equation (34). Indeed, although Equation ( 41) is different from the initial equation, namely (32), an adequate choice of stretching function leads the way back to it. We use a stretching function as the solution of the following eikonal equation

ρ ′ (z) 2 - 1 4ρ(z) 2 + R(0) ρ(z) = - 1 4z 2 + R(z) z , z ∈ C * . ( 42
)
Note that we call this equation the 'eikonal' equation because it plays the same role as the equation for φ if one seeks a solution of the wave equation (∆ -c -2 (x)∂ 2 t 2 )u = 0 under the form u(x, t) = a(x, k)e ik(φ(x)-t) . Note that (42

) implies ρ ′ (z) ρ(z) = ± 1 z 1-4zR(z)
1-4ρ(z)R(0) for z, ρ(z) non zero. As we choose ρ bounded in the neighborhood of 0, only the + sign is relevant. The equation rewrites

ρ(z) z ′ (z) = ρ(z) z F z, ρ(z) z , z ∈ C * , (43)
where F is defined by

F (z, a) = 4 [R(0)a -R(z)]
1 -4R(z)z 1 -4R(0)az + (1 -4R(0)az) .

We notice that the equation ( 42) is be defined also at z = 0 since F (0, a) makes sense, at least for small z and for bounded R since it guarantees that the square roots are defined without ambiguity. We now reintroduce R = R ν (• + X ν ) and consider the more general equation in the complex plane

σ ′ ν (z) = σ ν (z)F ν (z, σ ν (z)) , ρ ν (z) = zσ ν (z), z ∈ C, (44)
where F ν is defined by

F ν (z, a) = 4 [R ν (X ν )a -R ν (z + X ν )] 1 -4R ν (z + X ν )z 1 -4R ν (X ν )za + (1 -4R ν (X ν )za) .
Lemma 9 (Solution of the eikonal equation). Suppose x h is a local hybrid resonance. Under Assumptions 1, 2 and 3, consider Equation (44). Then there exists a constant Λ 7 satisfying 0 < Λ 7 ≤ Λ 6 such that:

• For all |ν| ≤ Λ 7 there exists a solution to Equation (42) on a maximal interval

I ν =]a ν , b ν [⊂] -Λ 3 , Λ 3 [ which contains the resonance x h .
• There exists a < 0 < b such that, for all |ν| < Λ 7 ,

x h ∈ I =]a, b[⊂ I ν .
• A solution of the eikonal equation ( 44) is ρ ν (z) for z ∈ B(0, Λ 7 ) defined

ρ ν (z) = z exp z 1 0 F ν (zt, σ ν (zt))dt . ( 45
)
Note that F is evaluated on the straight line from 0 to z

• This function is analytic in the ball z ∈ B(0, Λ 7 ) .
Proof. The function σ ν = ρν x is solution of the equation

σ ′ ν (z) = σ ν (z)F ν (z, σ ν (z)), z ∈ C. (46)
The value of F ν (0, a) = 2R ν (X ν ) (a -1) is a well defined (i.e. non singular, or bounded) real number. Equation (46) can be solved in the complex plane using a Theorem from Coddington-Levinson [START_REF] Coddington | Theory of ordinary differential equations[END_REF][theorem 8.1, page 34].

For simplicity consider the initial data σ ν (0) = 1 which yields σ ′ ν (0) = F ν (0, 1) = 0. This choice is arbitrary. Since σ ′ ν (0) = 0, one has the Taylor expansion ρ ν (z) = z + O ν (z 3 ) where the constant in O ν (z 3 ) ≤ C|z| 3 is uniform with respect to ν. All other coefficients of the Taylor expansion of the stretching function ρ ν can be easily computed from the integral representation in the complex plane (45) where the path of integration is the straight line. The analyticity is provided by the Coddington-Levinson theorem.

Thanks to the stretching function described in Lemma 9, it is now possible to explicit the solutions of Equation (41) where, in view of lemma 7 and equation (39), one needs to set

λ ν = 2 R ν ν ).
Definition 9. Suppose x h is a local hybrid resonance. Under Assumptions 1, 2 and 3, consider for all |ν| ≤ Λ 7 the stretching function ρ ν described in Lemma 9. A pair (U ν , V ν ) of independent solutions of Equation ( 41) is defined for all (ν, z)

∈ (-Λ 7 , Λ 7 ) × B(0, Λ 7 ) by                      U ν (z) = ρ ν (z) ρ ′ ν (z) J 0 λ ν ρ ν (z) , V ν (z) = ρ ν (z) ρ ′ ν (z) Y 0 λ ν ρ ν (z) -2 π log λν 2 J 0 λ ν ρ ν (z) = ρ ν (z) ρ ′ ν (z) T 0 λ ν ρ ν (z) + 1 π log (ρ ν (z)) J 0 λ ν ρ ν (z)
These functions are uniformly bounded with respect to ν.

Lemma 10. Suppose x h is a local hybrid resonance. Under Assumptions 1, 2 and 3, consider for all |ν| ≤ Λ 7 the stretching function ρ ν (45).

Then the functions U ν and V ν are bounded in a complex neighborhood of the resonance x h ∈ C, uniformly for small ν: there exists Λ 8 > 0 such that for all (ν, z) ∈ (-Λ 8 , Λ 8 ) × B(0, Λ 8 )

|U ν (z)| + |V ν (z)| ≤ C.
Moreover Λ 8 can be determined such that for all ν ∈ B(0, Λ 8 )

|ρ ′ ν (z)| ≥ δ > 0 in the same neighborhood.
Proof. The property on ρ ν is immediate since σ ν (0) = 1. The uniform boundedness of U ν stems from the boundedness of J 0 (36). Concerning V ν , the log λ ν term has been carefully removed, see (37), so that the boundedness is achieved even for vanishing λ ν : the remaining log z is controlled by the √ z contribution that comes from ρ ν (z).

It is now possible to apply the reverse shift.

Lemma 11. The functions x → U ν (x -X ν ) and x → V ν (x -X ν ) are solutions of the equation on the real line

d 2 dx 2 y(x) = - 1 4(x -X ν ) 2 + R ν (x) x -X ν + s ν (x -X ν ) y(x), x ∈ B(x h , Λ 8 ) ∩ R, (47)
where

s ν = (ρ ′ ν ) -1 2 ′′ (ρ ′ ν ) -1 2 .
Proof. By definition, all these fonctions are analytic in the ball B(x h , Λ 8 ) -{X ν } (that is except at X ν ) which is uniform with respect to ν. But, when ρ ν satisfies the eikonal relation (42) with R(z) = R ν (z + X ν ), the function z → U ν (z) (without the shift) is solution of (41). Therefore the identity (47) between analytic functions is true on B(x h , Λ 8 ) ∩ R, except at X ν for X ν ∈ R. So this relation is true everywhere by analytic continuation. The proof is ended.

Since X ν is non real for small non zero ν, this equation is valid also for small non zero ν.

with f (x) = -s ν (x -X ν ) (A ν (x)U ν (x -X ν ) + B ν (x)V ν (x -X ν ))
. One can solve now the linear system made of ( 54) and (55).

To compute the determinant of the linear system (54-55), consider the Wronskian of U ν and V ν . Since W (J0,Y0) (x) = 2 πx , elementary calculations yield

W (Uν ,Vν ) (x) = λ ν ρ ν (x -X ν ) ρ ′ ν (x -X ν ) 2 ρ ′ ν (x -X ν ) 2 ρ ν (x -X ν ) W (J0,Y0) λ ν ρ ν (x -X ν ) , so that W (Uν ,Vν ) (x) = 1 π .
By analytic continuation the determinant of the linear system (54-55) is also equal to the same value. Therefore the solution of the linear system (54-55) reads

A ′ ν (x) = -πs ν (x -X ν ) [A ν (x)U ν (x -X ν ) + B ν (x)V ν (x -X ν )] V ν (x -X ν ), B ′ ν (x) = πs ν (x -X ν ) [A ν (x)U ν (x -X ν ) + B ν (x)V ν (x -X ν )] U ν (x -X ν ).
After integration, it yields the representation formula (51). The boundedness of the kernel comes from the properties of s ν , U ν and V ν in Lemma 10. It completes the proof.

Under the same conditions, one can define the integral operator

K ν K ν (f )(x) = π x x * M ν (t)f (t)dt, so that the integral equation (51) reads C ν -K ν C ν = C ν (x *
), with a constant right hand side. Classically for this Volterra second type integral equation (see [START_REF] Tricomi | Integral equations[END_REF]), the solution is expressed with the resolvent kernel

Q ν [f ] = n≥1 K (n) ν (f ) under the form C ν (x) = C ν (x * ) + Q ν [C ν (x * )](x), = (Id + Q ν [1])(x)C ν (x * ), ∀x ∈ R, (56)
where C ν (x * ) is the initial condition. Foreseeing the limit process, the following results state the limit properties of ρ ν and M ν , and of additional quantities, as ν approaches zero.

Lemma 13. Suppose x h is a local hybrid resonance. Under Assumptions 1, 2 and 3, consider for all |ν| ≤ Λ 8 the stretching function ρ ν described in Lemma 9. One has that lim ν→0 ± ρ ν (z) = zσ 0 (z) on B(x h , Λ 8 ) where σ 0 is the solution of the integral equation ( 46) such that σ 0 (0) = 1. This limit function will naturally be called ρ 0 . Then the log and square root terms have limits which depend on the sign of ν: we set

log ρ 0 (x -x h ) ± = lim ν→0 ± log ρ ν (x -X ν ) = log (ρ 0 (x -x h )) for x h < x, log (-ρ 0 (x -x h )) ∓ iπs for x < x h , ( 57
)
where s = sign

i∂ν ǫ ν 11 (x) ∂xǫ ν 11 (x) |x=x h ,ν=0
, and

ρ 0 (x -x h ) ± = lim ν→0 ± ρ ν (x -X ν ) = ρ 0 (x -x h ) for x h < x, ∓is -ρ 0 (x -x h ) for x < x h , (58)
Proof. The claim on ρ 0 is evident. The limit of the log term comes from the principal value of the logarithm: log z = a(z) + ib(z) where a(z) = log |z| and b(z) ∈] -π, π]. And the limit of the square root term comes from the principal value of the complex square root:

√ z = |z|e i θ(z) 2
where θ(z) ∈] -π, π] is the argument of z.

Lemma 14. Suppose x h is a local hybrid resonance. Under Assumptions 1, 2 and 3, the matrix M ν defined in (52) has a limit in the following sense.

• M ν (x)

C 0 (x h -Λ8,x h +Λ8) -→ ν→0 + M + (x) • M ν (x) C 0 (x h -Λ8,x h +Λ8) -→ ν→0 - M -(x)
• The continuous kernels M + and M -are such that:

M + -M -= 0 vanishes on [x h , x h + Λ 8 ), while M + -M -= 0 is non identically zero on (x h -Λ 8 , x h ).
Consider (for simplicity) a Cauchy data C ν (x * ) which admits a finite (given) limit in ν in the representation (56) of the solution of integral equation. Then the functions A ν and B ν also converge in C 0 (x h -Λ 8 , x h + Λ 8 ) to smooth limit functions A ± 0 and B ± 0 . Proof. The continuous limits of the kernel M ν are given by application of Lemmas 13-14 since the matrix entries only depend on the functions U ν and V ν . The limit of functions A ν and B ν then directly stems from the limit of (56) where the kernel has a limit which is bounded. It ends the proof. Definition 10. We define three additional functions as follows:

             W ν 1 (x) = -c ν (x) ρν (x-Xν ) ρ ′ ν (x-Xν ) J 0 λ ν ρ ν (x -X ν ) = -c ν (x)U ν (x -X ν ) W ν 2 (x) = -c ν (x) ρν (x-Xν ) ρ ′ ν (x-Xν ) T 0 λ ν ρ ν (x -X ν ) + 1 π log(ρ ν (x -X ν ))J 0 λ ν ρ ν (x -X ν ) = -c ν (x)V ν (x -X ν ) , W ν 3 (x) = -c ν (x) ρν (x-Xν ) ρ ′ ν (x-Xν ) T 0 λ ν ρ ν (x -X ν ) . Since c ν (x)ρ ν (x -X ν ) is bounded, the functions W ν
1 and W ν 3 are uniformly bounded. On the other hand the function

W ν 2 (x) = W ν 3 (x) + 1 π log(ρ ν (x -X ν ))W ν 1 (x)
is not bounded due the log term.

Lemma 15. Suppose x h is a local hybrid resonance. Under Assumptions 1, 2 and 3, consider for all |ν| ≤ Λ 8 the stretching function ρ ν described in Lemma 9. One has

lim ν→0 ± W ν 1 (x) = W 1 (x) := d 0 (x)ρ 0 (x -x h ) ǫ 0 11 (x)ρ ′ 0 (x -x h ) J 0 (λ 0 ρ 0 (x -x h )), ∀x ∈ B(x h , Λ 8 ) ∩ R,
where the limit function W 1 is analytic on B(x h , Λ 8 ) ∩ R. Similarly

lim ν→0 ± W ν 3 (x) = W 3 (x) := d 0 (x)ρ 0 (x -x h ) ǫ 0 11 (x)ρ ′ 0 (x -x h ) T 0 (λ 0 ρ 0 (x -x h )), ∀x ∈ B(x h , Λ 8 ) ∩ R,
and

W 3 analytic on B(x h , Λ 8 ) ∩ R.
Proof. The point is to take the limit in the series (36) that defines J 0 . This is straightforward. Only even terms show up this series, so J 0 (λ 0 zσ 0 (z)) is analytic. For small x-x h the weight is the square root of d 0 (x)ρ0(x-x h )

ǫ 0 11 (x)ρ ′ 0 (x-x h )
which is well defined since d 0 (x)ρ0(x-x h )

ǫ 0 11 (x)ρ ′ 0 (x-x h ) ≈ d 0 (x h ) (ǫ 0
11 ) ′ (x h ) > 0 due to our normalization. So W 1 is analytic in the neighborhood of x h . Mutatis mutandis the proof is the same for the function W ν 3 . Considering the logarithmic part in Definition (37) of Y 0 , the situation is a little more involved concerning W ν 2 since the limits are dependent of the sign of ν. Lemma 16. Suppose x h is a local hybrid resonance. Under Assumptions 1, 2 and 3, consider for all |ν| ≤ Λ 8 the stretching function ρ ν described in Lemma 9. One has

lim ν→0 ± W ν 2 (x) = W ± 2 (x), ∀x ∈ B(x h , Λ 8 ) ∩ R,
where the limit functions are

W ± 2 = d 0 (x)ρ 0 (x -x h ) ǫ 0 11 (x)ρ ′ 0 (x -x h ) T 0 (λ 0 ρ 0 (x -x h )) + 1 π log (ρ 0 (x -x h )) ± J 0 λ 0 ρ 0 (x -x h ) (59)
The limit functions W + 2 and W - 2 are equal on the interval (x h , x h + Λ 8 ), but are different on the interval (x h -Λ 8 , x h ), where they satisfy the jump relation

W + 2 -W - 2 = -2iW 1 . (60)
Proof. Formula ( 59) is an immediate consequence of the definition of W ν 3 . The second relation comes from (57): we note that a similar relation has been proved in [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF].

Representation of the physical unknowns

One can now obtain meaningful representation formulas for the physical quantities we are interested in, being particularly careful about the various singularities encountered. Proposition 1. Suppose x h ∈ R is an isolated hybrid resonance, and assumptions 1, 2 and 3 hold. Consider for all |ν| ≤ Λ 8 the stretching function ρ ν described in Lemma 9. Consider for all ν a given reference point x * ∈ R close to x h in the ball of analyticity B(x h , Λ 8 ), as well as a Cauchy data C ν (x * ) which goes to a finite limit when ν goes to 0. Then the unique solution

(E ν 1 , E ν 2 , W ν ) of the regularized system (4)-(19) on B(x h , Λ 8 ) ∩ R is given by                            W ν = A ν W ν 1 + B ν W ν 3 + B ν 1 π log ρ ν (• -X ν )W ν 1 , E ν 2 = A ν iθǫ ν 12 d ν W ν 1 - ǫ ν 11 d ν d dx W ν 1 + B ν iθǫ ν 12 d ν W ν 3 - ǫ ν 11 d ν d dx W ν 3 + B ν 1 π log ρ ν (• -X ν ) iθǫ ν 12 d ν W ν 1 - ǫ ν 11 d ν d dx W ν 1 -B ν 1 π ρ ′ ν (• -X ν ) ρ ν (• -X ν ) ǫ ν 11 d ν W ν 1 , E ν 1 = A ν iθǫ ν 11 d ν W ν 1 + ǫ ν 12 d ν d dx W ν 1 + B ν iθǫ ν 11 d ν W ν 3 + ǫ ν 12 d ν d dx W ν 3 + B ν 1 π log ρ ν (• -X ν ) iθǫ ν 11 d ν W ν 1 + ǫ ν 12 d ν d dx W ν 1 + B ν 1 π ρ ′ ν (• -X ν ) ρ ν (• -X ν ) ǫ ν 12 (x) d ν W ν 1 . (61)
Proof. The expression of

W ν = √ -c ν (A ν U ν (• -X ν ) + B ν V ν (• -X ν )) is
E ν 2 = iθǫ ν 12 d ν W ν - ǫ ν 11 d ν d dx W ν . (62)
To compute d dx W ν we directly differentiate all terms in the expression already obtained for W ν (first line of (61)). We observe that the derivatives of A ν and B ν vanish since

A ′ ν U ν (• -x h ) + B ′ ν V ν (• -x h ) = 0, see (54). It yields d dx W ν = A ν d dx W ν 1 + B ν d dx W ν 3 + 1 π log ρ ν (• -X ν ) d dx W ν 1 + B ν 1 π ρ ′ ν (• -X ν ) ρ ν (• -X ν ) W ν 1 . (63)
It is then sufficient to plug this expression in (62) and to reorganize the sum to get the the representation formula for E ν 2 in the second line of (61). From [START_REF] Da | Stable coupling of the Yee scheme with a linear current model[END_REF], multiplying the first equality by ǫ ν 11 and the second equality by -ǫ ν 12 , adding, and using the symmetries of the dielectric tensor, one gets

E ν 1 = iθǫ ν 11 d ν W ν + ǫ ν 12 d ν d dx W ν . (64)
Even if very simple, this algebra seems to be important since various cancellations of potential singular terms have been performed. It is then sufficient to plug the representation formulas for W ν and d dx W ν in (64) to obtain the third line of the claim.

At inspection of the representation formulas (61), it is clear that the convergence with respect to ν is not the same for E ν 2 and W ν on the one hand, and for E ν 1 on the other hand. Indeed the most singular term in E ν 2 and W ν is the logarithm log ρ ν (• -X ν ): the last term in E ν 2 is non singular since

ǫ ν 11 ρν (•-Xν )
is the ratio of two singular terms; in terms of singular behavior these to terms cancel each other. So these terms all terms are bounded in L p loc for p < ∞ and pass to the limit point wise except at the singularity x h = lim ν→0 X ν . This result was already obtained for the slab geometry with a completely different technique in [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF][proposition 5.14] and is generalized here. So we state without detail the representation formulas for the limits.

Proposition 2. Suppose x h is a local hybrid resonance, and assumptions 1, 4 and 3 hold. Consider for all |ν| ≤ Λ 8 the stretching function ρ ν described in Lemma 9, and C ν (x * ) which goes to a finite limit when ν goes to 0. Consider the unique solution (E ν 1 , E ν 2 , W ν ) of the regularized system (4) on B(x h , Λ 8 ). Then for all 1 ≤ p < ∞ the L p limit as ν → 0 of W ν and E ν 2 are

W ± = A ± 0 W 1 + B ± 0 (x)W 3 + B ± 0 1 π log ρ 0 (• -x h ) ± W 1 ( 65
)
and

E ± 2 = A ± 0 iθǫ 0 12 d 0 W 1 - ǫ 0 11 d 0 d dx W 1 + B ± 0 iθǫ 0 12 d 0 W 3 - ǫ 0 11 d 0 d dx W 3 + B ± 0 1 π log ρ 0 (• -x h ) ± iθǫ 0 12 d 0 W 1 - ǫ 0 11 d0 d dx W 1 -B ± 0 1 π ρ ′ 0 (• -x h ) ρ 0 (• -x h ) ǫ 0 11 d 0 W 1 (66)
where log ρ 0 (• -x h ) ± is defined in (57). The limits hold also on pointwise in R -{x h } ∩ B(x h , Λ 8 ).

The situation is different for the E ν 1 component (61) since the division by ρ ν (• -X ν ) in the last term yields to a singularity as ν approaches zero. This behavior is the same that already demonstrated in [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF] using an singular integral equation technique. The difference is that we have now an explicit representation of this singular behavior for small ν. To express the limit of the singular term, the more efficient way is to use principal value and Dirac mass. Since the result is essentially the same as in [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF] and all calculations are now evident starting from the representation formula (61) we state the result without details of the proof. Theorem 1. Suppose x h is a local hybrid resonance, and assumptions 1, 4 and 3 hold. Consider for all |ν| ≤ Λ 8 the stretching function ρ ν described in Lemma 9, and C ν (x * ) which goes to a finite limit when ν → 0. Consider the unique solution (E ν 1 , E ν 2 , W ν ) of the regularized system (4) on B(x h , Λ 8 ). Then the limit of

E 1 ν is on B(x h , Λ 8 ) E ± 1 = A ± 0 iθǫ 0 11 d 0 W 1 + ǫ 0 12 d 0 d dx W 1 + B ± 0 iθǫ 0 11 d 0 W 3 + ǫ 0 12 d 0 d dx W 3 + B ± 0 1 π log ρ 0 (• -x h ) iθǫ 0 11 d 0 W 1 + ǫ 0 12 d 0 d dx W 1 +iB ± 0 (0) ǫ 0 12 (0) d 0 D(• -x h ) + P.V. B ± 0 1 π ρ ′ 0 (•-x h ) ρ0(•-x h ) ǫ 0 12 (x) d 0 W 1 ,
where D is the Dirac mass and the principal value is defined by <

P.V.(α), φ >= lim τ →0 + x ∈[x h -τ,x h +τ ] α(x)φ(x)dx.
Of course the right hand side terms on the first line converge also in L p loc as in the previous proposition, and the point wise limit holds away from the hybrid singularity x h .

Limit heating term

The heating was defined in the formula [START_REF] Silva | A Numerical Study of Forward-and Backscattering Signatures on Doppler-Reflectometry Signals[END_REF]. For a < b in B(x h , Λ 8 ) ∩ R * and ν ∈ (-Λ 8 , Λ 8 ) we note

Q ν (a, b) = ℑ b a (ǫ ν (x)E ν (x)) • E ν (x)dx = -ℑ W ν (b)E ν 2 (b) + ℑ W ν (a)E ν 2 (a) (67)
where (E ν 1 , E ν 2 , W ν ) is the regularized solution system (4) on B(x h , Λ 8 ). The limit value for vanishing ν is the resonant heating and can be characterized.

Theorem 2 (Resonant heating). Suppose x h is a local hybrid resonance, and assumptions 1, 4 and 3 hold. Consider for all |ν| ≤ Λ 8 the stretching function ρ ν described in Lemma 9, and C ν (x * ) which goes to a finite limit when ν goes to 0. One can pass to the limit for (a, b)

∈ B(x h , Λ 8 ) 2 ∩ R * as follows. i) For (a -x h )(b -x h ) > 0 then lim ν→0 Q ν (a, b) = 0. ii) If a < x h < b, then lim ν→0 + Q ν (a, b) = 1 π |B 0 (x h )| 2 ǫ 0 12 (0) 2 sign ∂ ν ǫ 0 11 . (68)
Proof. There are many possibilities to compute this limit value from the previous representation formulas.

First case) This is a direct consequence on the fact that a and b are on the same side of x h , that the solution is smooth away from x h and that ℑ ǫ 0 (x)X • X = 0 for all complex vector X since ǫ 0 (x) is hermitian.

Second case) It is sufficient to decompose the electric field in two parts, a regular part R ν and a singular part S ν , such that R ν is regular enough, for example bounded in L p (-Λ 8 , Λ 8 ) uniformly with respect to ν and with a point wise limit for x = x h and the singular part S ν is such that (x -X ν )S ν is bounded in L ∞ (-Λ 8 , Λ 8 ). It is then clear that one can choose

S ν (x) = B ν 1 π ρ ′ ν (• -X ν ) ρ ν (• -X ν ) ǫ ν 12 d ν W ν 1 , 0 .
One has thus

Q(a, b) = ℑ b a (ǫ ν (x)R ν (x)) • R ν (x)dx + (ǫ ν (x)S ν (x)) • R ν (x)dx+ (ǫ ν (x)R ν (x)) • S ν (x)dx + (ǫ ν (x)S ν (x)) • S ν (x) dx One checks that ℑ b a (ǫ ν (x)S ν (x)) • R ν (x)dx = ℑ b a ǫ ν (x) 11 S ν 1 (x)R ν 1 (x)dx and that ℑ b a (ǫ ν (x)R ν (x)) • S ν (x)dx = ℑ b a (R ν (x)) • ǫ ν (x)S ν (x)dx = ℑ b a (R ν 11 (x)) • ǫ ν 11 (x)S ν 1 (x)dx.
These two terms converge to 0 using the dominated convergence theorem as ν → 0 thanks to the regularity of (x -X ν )S ν 1 (x) and of (x -X ν )S ν 1 (x) in the neighborhood of x h . Far better is, of course, the term ℑ b a (ǫ ν (x)R ν (x)) • R ν (x)dx, which limit is also 0. Let us study the remaining term which is

J = b a (ǫ ν (x)S ν (x)) • S ν (x) = b a ǫ ν 11 (x) B ν (x) 1 π ρ ′ ν (x -X ν ) ρ ν (x -X ν ) ǫ ν 12 (x) d ν (x) W ν 1 (x) 2 dx.
One denotes by z ν (x) =

ǫ ν 11 (x)
x-Xν and by

K ν (x) = (x -X ν )B ν (x) 1 π ρ ′ ν (x-Xν ) ρν (x-Xν ) ǫ ν 12 (x) d ν (x) W ν 1 (x)
. These two terms are regular, hence

J = b a z ν (x)|K ν (x)| 2 x -X ν |x -X ν | 2 dx. One introduces J 0 = b a z ν (X ν )|K ν (X ν )| 2 x-Xν |x-Xν | 2 dx.
One checks that there exists a regular function

H ν (x) such that z ν (x)|K ν (x)| 2 -z ν (X ν )|K ν (X ν )| 2 = (x -X ν )H ν (x). We thus deduce that J -J 0 = b a H ν (x) (x -X ν ) 2 |x -X ν | 2 dx.
As the limit exists in L 1 , we use the dominated convergence theorem to deduce that the limit of J -J 0 is equal to

J 0 = z ν (X ν )|K ν (X ν )| 2 b-ℜXν ℑXν a-ℜXν ℑXν t -i t 2 + 1 dt.
We thus notice that

J 0 = z ν (X ν )|K ν (X ν )| 2 [A ν -iB ν ]
where A ν is given thanks to 1 2 ln(1 + t 2 ) and B ν = tan -1 ( b-ℜXν ℑXν ) -tan -1 ( a-ℜXν ℑXν ). One observes that the limit of z ν (X ν )) when ν → 0 is real, equal to ∂ x ǫ 0 11 (x h ) (because the limit of z ν is equal to

ǫ 0 11 (x)
x-x h ), and the limit of

ǫ 0 11 (x) x-x h when x → x h is ∂ x ǫ 0 11 (x h ), real. One has A ν = 1 2 ln (b -ℜX ν ) 2 + (ℑX ν ) 2 (a -ℜX ν ) 2 + (ℑX ν ) 2 → 1 2 ln (b -x h ) 2 (a -x h ) 2 ∈ R.
When a < x h < b, the limit of a-ℜXν ℑXν is -∞ when ℑX ν goes to 0 by positive values, and the limit of a-ℜXν ℑXν is +∞ when ℑX ν goes to 0 by negative values. We thus deduce

B ν -→ πsign(ℑX ν ).
Hence one deduces that the limit of Im J 0 is -π∂ x ǫ 0 11 (x h )|B ± 0 (0)W 1 (0)

ǫ 0 12 (x h ) πd 0 (x h ) | 2 . Using d 0 (x h ) = (ǫ 0 12 (x h )) 2 , one obtains that ℑJ -→ -sign ℑX ν ∂ x ǫ 0 11 (x h ) π|ǫ 0 12 (x h )| 2 |B ± 0 (0)W 1 (0)| 2 . Using finally that |W 1 (0)| 2 = |ǫ 0 12 (x h )| 2 |∂xǫ 0 11 (x h )| , one gets ℑJ -→ -sign ℑX ν |B ± 0 (0)| 2 as ν tends to zero.
The proof is ended.

An interesting and somewhat counter intuitive corollary is the following.

Corollary 1. Consider any dielectric tensor satisfying (4-24) but not necessarily with the exact form (3). Assume ǫ 0 is a smooth hermitian matrix, and assume x h = 0 is an hybrid resonance. Define the local dissipation tensor

D = -i (∂ ν ǫ) 0 (0) = d 1 d 2 -d 2 d 1 provided the first coefficient is positive d 1 > 0.
Then the value of the resonant heating is independent of D. The same for the pointwise limits of the electric, of magnetic fields and of the numerical vale of the resonant heating.

Proof. It is sufficient to realize that the major assumption that was made, that is (20-13) is independent of the exact value of d 1 > 0, and that the other coefficients in D do not show up in the proof of the previous theorem. This is also clear in view of the value of the resonant heating (68).

An interesting consequence is that can replace the initial dielectric tensor (3) with the linear approximation ǫ 0 + iνD with D defined in [START_REF] Chen | Amplification and Absorption of Electromagnetic Waves in Overdense Plasmas[END_REF]. If one is interested only in the limit value, which is a reasonable assumption for fusion plasmas where ν ≈ 10 -7 may be encountered, this is a valid assumption.

4 Numerical illustrations and analytic solutions

Numerics

We show numerical results which illustrate some of the theoretical results. These results have been computed with the Matlab solver developed by Lise-Marie Imbert-Gérard during her PhD [START_REF] Imbert-Gérard | Analyse mathématique et numérique de problémes d'ondes apparaissant dans les plasmas magnétiques[END_REF]. The code solves the system under the form [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF] with the ode23 subroutine of Matlab. This routine is adapted to stiff problems. We use three different dielectric tensors that all have the same limit for ν = 0. The three dielectric tensors are denoted by three different "dissipation" tensors. The first one denoted as D 0 means that we use the exact formula (3). The second one D 1 = 1 0 0 1 means that we use the linear approximation ǫ 0 + iνD 1 . One can notice that D 1 = I corresponds to the limiting absorption principle that was studied in [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF]. The last

D 2 = D * 2 = 1 -2i 2i 5
is a hermitian and positive matrix as is required on the physical basis (see (6) for example). These three methods satisfy the main hypothesis of corollary 1 that is (D i ) 11 > 0 for i = 0, 1, 2.

Once the dissipation tensor is set, the matrix M ν (x, θ) is constructed, and the problem solved numerically in the interval [-5, 5] with Cauchy data E 2 (5) = 1 and W (5) = 0. The other parameters are chosen so that x h = 0 is an isolated hybrid resonance, that x < 0 is a propagating and that < 0 is non propagative. This is clearly visible in table 1. In figure 3 we display the real part of E ν x , E ν y , W ν for four different values of ν, and for the linear models with D 1 and D 2 . We observe that even if the results for ν = 0.5 are very different, the functions converge numerically to the same limit for ν = 0.5 × 10 -3 . The next figure 4 confirms this analysis for the imaginary of the electromagnetic field. The final figure 5 shows the same results but computed with the "real" dissipation tensor. One clearly observes the same limit as in figures 3 and 4, perhaps with smaller oscillations for large value ν. ν .5 .5 × 10 -1 .5 × 10 -2 .5 × 10 -3 .5 × 10 -4 .5 × 10 As predicted by the theory, the limit resonant heating in the last column is independent of the dissipation tensor. One observes the convergence to the same limit. One observes the convergence to the same limit. • D 0 (x) vanishes at points x hi such that A 0 (x hi ), B 0 (x hi ), C 0 (x hi ) do not vanish and (A 2 0 + B 0 C 0 )(x hi ) = 0 • i∂ ν D ν (x hi )| ν=0 ∈ R = 0 which are the unique conditions needed for the theoretical set-up of our main result. This coefficient D ν is analogous to ǫ ν 11 . For simplicity we describe a two species case, one is electron and one is ions and just give the main ideas. To obtain ( 69 = Zme mi . The traditional approach is to calculate v in terms of E (which introduces a singularity at x such that ω i c (x) = ω), then to deduce j in terms of E (which introduces a singularity at x such that ω e c (x) = ω) and finally to replace j in Maxwell equations. This structure is not the natural one in the sense that the dielectric tensor that one obtains has singularities when ν goes to 0. Instead of using this approach, we seek the TE field (E 2 (x), B 3 (x) := W (x))e iθy , solution of

c 2 W ′ = iωE 2 -4πj 2 E ′ 2 = iθE 1 + iωW
We have thus to obtain (j 2 , E 1 ). It can be checked they are solution of              (-iω + ν)j 2 + ω e c j 1 - . One is thus left with finding x hi such that ω = ω e c (x hi )(X * (x hi ))

1 2 . Other formulas can be sought for.

a 2 ν

 2 cν , b ν and aν cν ′ term all behave as O((ǫ ν ) -1 ) at the resonance.

b a H 0

 0 (x)dx. It is real hence its imaginary part is zero. Consider now J 0 . Use x-Xν |x-Xν | 2 = x-ℜXν -iℑXν (x-ℜXν ) 2 +(ℑXν ) 2 . The change of variable t = x-ℜXν ℑXν yields

Figure 3 :

 3 Figure 3: Real part, D 1 on the left, D 2 on the right, from top to bottom: ν = .5, .5 • 10 -1 , .5 • 10 -2 , .5 • 10 -3 .One observes the convergence to the same limit.

Figure 4 :

 4 Figure 4: Imaginary part, D 1 on the left, D 2 on the right, from top to bottom: ν = .5, .5 • 10 -1 , .5 • 10 -2 , .5 • 10 -3 .One observes the convergence to the same limit.

Figure 5 :

 5 Figure 5: D 0 , real part on the left, imaginary part on the right, from top to bottom: ν = .5, .5 • 10 -1 , .5 • 10 -2 , .5 • 10 -3 . For the smallest value of ν, the results are almost the same as in figures 3 and 4 computed with the linear models D 1 and D 2 .

  ), one can start from the Maxwell equations∇ ∧ E = iωB, c 2 ∇ ∧ B = -iωE + 4πjwith the so-called fluid equations for the current j (the external magnetic field B 0 is B 0 (x), b, b unit vector) :(-iω + ν)j = eB 0 (x) m e j ∧ b -ω 2 p 4π (E + B 0 (x)v ∧ b)where the ion velocity v is given by (-iω + ν)v = Ze mi (E + B 0 (x)v ∧ b).The physical relevant parameters are ω i c = ZeB0 mi , ω e c = eB0 me ,

2 p4π ( 1 + 2 p4π ( 1 + p)B 0 v 2 + ω 2 p4π ( 1 +) 2 ,

 2121212 p)E 2 -ω e c j 2 + (-iω + ν)j 1 + ω p)E 1 = 0 (-iω + ν)v 1 -ω i c v 2 -Ze mi E 1 = 0 ω i c v 2 + (-iω + ν)v 1 = Ze mi E 2 4πj 1 -iωE 1 = c 2 iθWTreating this system globally leads to a determinant D ν :D ν (x) = iω((ω i c ) 2 + (iω -ν) 2 )((ω e c ) 2 -(iω -ν) 2 ) + (iω -ν)((iω -ν) 2 + ω i c ω e c )ω 2 p (1 + p). One has D0(x) iω = (p 2 (ω e c ) 2 -ω 2 )((ω e c ) 2 + ω 2 ) + (1 + p)ω 2 p (p(ω e c ) 2 -ω 2). In this case, for all x, there exist a unique value of ω for which there is an hybrid resonance such thatD 0 (x) = 0. If one denotes X = ( ω ω e c X is solution of -X 2 + (p + 1)(p -1 -X = X * (x), equivalent to ω = ω e c (x)(X * (x))1 2

  immediate from Lemma 12, since it gives the general solution (49) of the corresponding second order equation. It yields the first identity. Concerning the second identity, one can start from E ν 2 = aν cν W ν + 1

	(25) one has 1 cν = -	ǫ ν 11 d ν and aν cν =	iθǫ ν 12 d	cν	d dx W ν by means of (11) and (26). Using

ν . So one can write

Table 1 :

 1 Values of the heating calculated with formula (67) with different ν and different dissipation tensors.

	-5

* Lise-Marie Imbert-Gérard and Bruno Després acknowledge the support of ANR under contract ANR-12-BS01-0006-01. Moreover this work was carried out within the framework of the European Fusion Development Agreement and the French Research Federation for Fusion Studies. It is supported by the European Communities under the contract of Association between Euratom and CEA. The views and opinions expressed herein do not necessarily reflect those of the European Commission.

General solution with a Duhamel's principle and limit process

In order to go back to the equation with a varying coefficient, let us now consider the original problem (32). One can rewrite this equation as: for all x ∈ R

The last term is a perturbation with respect to Equation (47). As usual for such problems, this term is treated by means of the Duhamel principle where we will make major profit of the fact that the fundamental solutions of the singular equation (47) are bounded.

Lemma 12. Suppose x h ∈ R is an isolated hybrid resonance, and assumptions 1, 2 and 3 hold. Consider for all ν a given reference point x * ∈ R close to x h in the ball of analyticity B(x h , Λ 8 ). Then the function

is the solution of the integral equation

where the matrix

As usual for such Volterra integral equations, the initial point x * ∈ R can be chosen arbitrarily.

Proof. This is the standard procedure of variation of parameters. Consider the following equation:

with f (x) = -s ν (x -X ν )y. The solution y can be expressed as a combination of U ν (x -X ν ) and V ν (x -X ν ), with appropriate coefficients A ν (x) and B ν (x)

To construct the coefficients A ν (x) and B ν (x), we first assume that

So the first derivative of y is y

The second derivative reads

Using (47), one gets

Since y is solution of (53), one gets

Two analytic solutions

The first case of interest (for which s = 0) is ρ ν (x) =

x 1+l -1 0 x , l 0 being an arbitrary length. Choose arbitrarily X ν , (such as X ν = -iν). Consider:

where λ, l 0 are arbitrary constants.

Lemma 17. The family of solutions W ν and E ν 2 of ( 11), with the above coefficients of M ν defined in ( 12) is given by

2 is a polynomial of degree 1, hence s = 0. We construct coefficients a ν , b ν , c ν according to the form of the solution. Introduce

and the choice of a ν , b ν , c ν satisfy this equality. Note in particular that the relation

ν f = 0. The second case is the case where the solution is singular, namely

(we consider the solution as a combination of Bessel functions), and ρ ν (x) = x(1 + l -2 0 x 2 ). One may choose in this case

With a choice of the arbitrary constants K, C 0 and l 0 , and with a convenient choice of X ν , one has Lemma 18. All solutions of

A short conclusion

All our numerical results confirm the singular solution predicted by the theoretical resuls. It allows us now to have a comprehensive understanding of the local structure of the hybrid singular solutions. With a careful expression of C ν (x * ) through E ν 2 (x * ) and W ν (x * ) and using the methods developed in [START_REF] Lafitte | High-frequency stability of multidimensional ZND detonations[END_REF], one can, as in [START_REF] Després | Hybrid resonance of Maxwell's equations in slab geometry[END_REF], connect this local structure near the singularity to the condition at infinity.

A Proof of Lemma 6

Proof. This function is a Laurent series by definition. Hence we will show that there exists Λ 6 > 0 such that R ν (x) is uniformly bounded, i.e. bounded for x ∈ B(x h , Λ 6 ) uniformly for ν ∈ B(0, Λ 6 ). And such a Λ 6 can be replaced by Λ 3 in case this one was smaller. This will prove that it is an analytic function with a uniform radius of convergence. The function R ν is the product of x -X ν times a contribution which is the sum of four terms. The first term is

. The product by x -X ν is bounded and admits a finite limit for x → X ν . The second term depends on aν cν which is locally bounded, so does not yield any difficulty either: its contribution to R ν is bounded. The third term can be written as

Considering the bounded function

•-Xν where the function τ ν is uniformly bounded in the sense that |τ ν (x)| ≤ C, i.e. bounded for x ∈ B(x h , Λ 6 ) uniformly for ν ∈ B(0, Λ 6 ). It shows that the most singular part of the two last terms cancel in the definition of R ν . The exact value of R ν (X ν ) may be computed as follows using d ν defined in [START_REF] Lafitte | High-frequency stability of multidimensional ZND detonations[END_REF]. From aν cν = iθ

Moreover, noticing that k ν = -d ν x-Xν ǫ ν

11

, and using the l'Hopital rule with ǫ

(ǫ ν 11 ) ′′ (ǫ ν 11 ) ′ (X ν ).

B Multi species

Our aim here is to show that our method can be used to study the resonances in the case of multi-species. For simplicity, let us consider the case of the electron-ion system. After elementary calculations based on the fundamental equations that one may find in [START_REF] Stix | The Theory of Plasma Waves[END_REF][START_REF] Swanson | Plasma Waves[END_REF], one obtains for example the generalization of (26)

Here D ν satisfies