Hélène Frisch
email: frisch@oca.eu

Exact solutions for radiative transfer with partial frequency redistribution

Keywords: radiative transfer-scattering-stars, atmospheres-methods, analytical 1. Introduction

 for a partial frequency redistribution model of the formation of spectral lines consisting in a linear combination of frequency coherent and fully incoherent scattering. The method of solution is based on an eigenfunction expansion of the radiation field, leading to two singular integral equations with Cauchy or Cauchy-type kernels, that have to be solved one after the other. Aims. We reconsider this problem, using as starting point the integral formulation of the radiative transfer equation, where the terms involving the coupling between the two scattering mechanisms are clearly displayed, as well as the primary source of photons. Methods. With an inverse Laplace transform, we recover the singular integral equations previously established and, with Hilbert transforms as in the previous work, recast them as boundary value problems in the complex plane. Their solutions are presented in detail for an infinite and a semi-infinite medium. The coupling terms are carefully analyzed and consistency with either the coherent or the incoherent limit is systematically checked. Results. We recover the important result of the previous work that an exact solution exists for an infinite medium, whereas for a semi-infinite medium, which requires the introduction of half-space auxiliary functions, the solution is given by a Fredholm integral equation to be solved numerically. The solutions of the singular integral equations are used to construct explicit expressions providing the radiation field for an arbitrary primary source and for the Green function. An explicit expression is given for the radiation field emerging from a semi-infinite medium.

 (1962)to analyze the formation of these lines. It takes into account the spontaneous frequency coherent deexcitation of the upper level. When incorporated into a radiative transfer equation, it leads to somewhat cumbersome numerical work. Several approximations consisting in a linear combination of monochromatic scattering and complete frequency redistribution have subsequently been proposed in the literature (see [START_REF] Hubeny | Theory of stellar atmospheres[END_REF] for a comprehensive review). A linear combination of monochromatic scattering and complete frequency redistribution is also a very crude model for spectral lines formed between two excited levels [START_REF] Jefferies | Spectral Line Formation[END_REF][START_REF] Frisch | Non-LTE transfer V. The asymptotics of partial redistribution[END_REF][START_REF] Hubeny | Theory of stellar atmospheres[END_REF]Frisch 2022, p. 252). In different domains, this same model was introduced by [START_REF] Corngold | Analysis of pulsed-neutron experiments in moderators via a simple model[END_REF] to analyze the decay of a neutron pulse inside a moderator, and by [START_REF] Yengibarian | On a problem of multiple resonance scattering of gamma-ray quanta[END_REF] to study the resonance scattering of gamma-ray quanta. It must be said that for resonance lines, models combining complete frequency redistribution and monochromatic scattering fail at reproducing some specific features, such as the shape of the lines wings or the linear polarization created by multiple scattering. [START_REF] Hummer | Non-coherent scattering I. The redistribution functions with Doppler broadening[END_REF][START_REF] Faurobert | Linear polarization of resonance lines in the absence of magnetic files II Semi-infinite atmospheres[END_REF][START_REF] Hubeny | Theory of stellar atmospheres[END_REF]. The reason is that they ignore the broadening by thermal velocities of the monochromatic component, which dominates in the lines wings.

A&A proofs: manuscript no. mixture-aa-general are exact solutions for an infinite medium but not for a semiinfinite one. In the latter case, the solution is provided by a Fredholm integral equation, which has to be solved numerically.

A method for constructing exact solutions is also presented in [START_REF] Yengibarian | On a problem of multiple resonance scattering of gamma-ray quanta[END_REF], but the results are not as explicit as those described in [START_REF] Hemsch | Radiative transfer with partially coherent scattering[END_REF]. Because of the interplay between monochromatic scattering and complete frequency redistribution, the solution is of course not a linear combination of a monochromatic scattering and a complete frequency redistribution solution. It has to be constructed in two successive steps. The first one consists in solving a monochromatic scattering problem. The integration over frequency of its solution provides a generalized complete frequency redistribution, involving only frequency integrated functions, which can be solved exactly for a infinite medium but not for a semi-infinite one.

We consider in this article the same partial frequency redistribution problem, but address it with a new method of solution. In [START_REF] Hemsch | Linear transport theory with partly incoherent scattering[END_REF] and [START_REF] Hemsch | Radiative transfer with partially coherent scattering[END_REF], exact solutions are derived from an eigenfunction expansion of the radiation field. This method, introduced by Case (1960), has been widely applied to several neutron transport problems and some astrophysical ones (e.g., [START_REF] Case | Linear Transport Theory[END_REF][START_REF] Mc Cormick | Singular eigenfunction expansions in neutron transport theory[END_REF][START_REF] Duderstadt | Transport Theory[END_REF].

It leads to singular integral equations with Cauchy-type kernels. The latter are then solved by the standard transformation into boundary value problems in the complex plane (e.g., [START_REF] Muskhelishvili | Singular Integral Equations[END_REF][START_REF] Gakhov | Boundary Value Problems[END_REF][START_REF] Pogorzelski | Integral Equations and their Applications[END_REF][START_REF] Ablowitz | Complex Variables, Introduction and Applications[END_REF]. In [START_REF] Yengibarian | On a problem of multiple resonance scattering of gamma-ray quanta[END_REF] the radiative transfer equation is first transformed into convolution-type integral equations and the solution derived by several manipulations of these integrals, inspired by the resolvent method [START_REF] Ivanov | Resolvent method : exact solutions of half-space transport problems by elementary means[END_REF]). In the present article, we start also from convolution-type integral equations. The application of an inverse Laplace transform provides singular integral equations with Cauchy-type kernels. They are very similar to those derived in [START_REF] Hemsch | Radiative transfer with partially coherent scattering[END_REF], and solved also by transformations into boundary value problems in the complex plane. Our results fully confirm those of M. Hemsch and J. Ferziger.

The organization of this work is as follows. The radiative transfer and associated integral equations are presented in Section 2. Section 3 is devoted to the monochromatic dispersion function, which plays a critical role in the solution of the monochromatic scattering problem. Solutions of the radiative transfer problem are constructed for an infinite medium and a semi-infinite one, respectively in Sections 4 and 5. It is assumed for simplicity, but without loss of generality, that the ratio of monochromatic scattering to complete frequency redistribution, is a constant (independent of space and frequency). Solutions are given for the Green function and for an arbitrary primary source of photons. In Section 5 we also show how to calculate the emergent radiation field and indicate the changes are are needed when the ratio of monochromatic scattering to complete frequency redistribution depends on the frequency. Two Appendices contain technical material. The first one discusses in detail the properties of a generalized dispersion function, associated to the generalized complete frequency redistribution problem of the second step.

The radiative transfer equation and associated integral equations

We give here the radiative transfer equation corresponding to a combination of coherent and fully incoherent scattering, then in-tegral equations that can be derived from it, namely convolution-116 type integral equations in the physical space and singular integral 117 equations used to seek for exact solutions. For a one-dimensional medium, the radiative transfer equation 120 may be written as

121 µ ∂I ∂τ (τ, x, µ) = ϕ(x)[I(τ, x, µ) -S (τ, x)], (1)
where I(τ, x, µ) is the specific intensity of the radiation field. 122 Here µ is the direction cosine of a ray, τ the line integrated optical 123 depth, x the frequency variable measured in Doppler-width units 124 with 0 at line center. The domains of definition of these variables 125

are µ ∈ [-1, +1], τ ∈] -∞, +∞[, x ∈] -∞, +∞[ 1 .
S (τ, x) = (1 -ǫ)[bJ(τ, x) + (1 -b) J(τ)] + Q * (τ), (2)
where Q * (τ) is a given primary source of photons,

131 J(τ, x) = 1 2 +1 -1 I(τ, x, µ) dµ, J(τ) = +∞ -∞ J(τ, x)ϕ(x) dx. (3)
The parameter ǫ is a positive constant smaller than unity, giv-132 ing the destruction probability per scattering. The coefficient 133 b ∈ [0, 1] is the branching ratio between monochromatic scat-134 tering and complete frequency redistribution, with b = 0, in 135 the limit of complete frequency redistribution and b = 1 in 136 the limit of monochromatic scattering. In [START_REF] Hemsch | Linear transport theory with partly incoherent scattering[END_REF] and 137 [START_REF] Hemsch | Radiative transfer with partially coherent scattering[END_REF], b is a function of x. Here we assume 138 that b is a constant. This assumption does not affect the main 139 results, as we show in Section 5.4. A frequency dependent b is 140 found in all approximations to the partial redistribution function 141 for resonance lines, whereas a constant b is a very simplified 142 model for the partial frequency redistribution function, relevant 143 to spectral lines formed between two excited levels (Hummer 144 1962;[START_REF] Hubeny | Theory of stellar atmospheres[END_REF]Frisch 2022, p. 569).

145

In [START_REF] Hemsch | Radiative transfer with partially coherent scattering[END_REF] the solution of Eq. ( 1) is 146 searched for in the form of an eigenfunction expansion :

147 I(τ, x, µ) = k Φ(k, x, µ) e -τ/k . ( 4
)
It can be viewed as a continuous version of the Wick-148 Chandrasakhar discrete ordinates method (Kourganoff 1950, 149 [START_REF] Chandrasekhar | Radiative Transfer[END_REF]. When inserted into Eq. ( 1), this expan-150 sion leads to a singular integral equation with a Cauchy kernel. 151

Here we make a similar assumption, namely that J(τ, x) can be 152 written as a Laplace transform (see Eq. ( 12)). Associated to an 153 integral equation for J(τ, x) written in the next Section, it leads 154 to a singular integral equation very similar to that given by the 155 eigenfunction expansion method.

ξ = ϕ(x), η = 1/µ, (5)
and the single scattering albedo,

ω ≡ 1 -ǫ. (6)
The variable ξ is positive and its range of variation is [0, ϕ 0 ], with ϕ 0 = ϕ(0). Associated to this change of variable we introduce the 164 new functions J(τ, ξ) = J(τ, x) and J(τ), which is defined by

165 J(τ) = J(τ) = +∞ -∞ ϕ(x)J(τ, x) dx = ϕ 0 0 W(ξ)J(τ, ξ) dξ, (7)
where 166

W(ξ) = 2| dx dξ | ξ, ϕ 0 = ϕ(0). ( 8
)
The normalization of ϕ(x) to one implies

167 ϕ 0 0 W(ξ) dξ = 1. (9)
The standard method for transforming the radiative transfer

J(τ, ξ) = ω ξ +∞ -∞ [bJ(τ ′ , ξ))] K M (|τ ′ -τ|ξ) dτ ′ + ξ +∞ -∞ [Q * (τ ′ ) + ω(1 -b) J(τ ′ )] K M (|τ ′ -τ|ξ) dτ ′ . ( 10
)
where for an infinite medium. For the semi-infinite medium, treated in 187 Section 5, it takes exactly the same form, the integration over τ 188 being then from 0 to ∞. The transformation of the convolution integral equation for J(τ, ξ) into a singular integral equation with a Cauchy-kernel is based on the assumption that J(τ, ξ) can be written as a Laplace transform, in other terms that we can write

J(τ, ξ) =            ∞ 0 j(ν, ξ) e -ντ dν τ > 0, - 0 -∞ j(ν, ξ) e -ντ dν τ < 0. ( 12
)
Here ν is real and j(ν, ξ) can be a regular function or a distribution. A similar representation is used for J(τ) and Q * (τ). Their inverse Laplace transforms are denoted (ν) and q * (ν). An integration over ξ of Eq. ( 12) shows that

(ν) = ϕ 0 0 W(ξ) j(ν, ξ) dξ. ( 13
)
We also introduce

p * (ν) ≡ ω(1 -b) (ν) + q * (ν). ( 14
)
The transformation of Eq. ( 10) into a singular integral equation with a Cauchy-type kernel requires some algebra introduced in [START_REF] Frisch | A method of Cauchy integral equations for noncoherent transfer in half-space[END_REF] for complete frequency redistribution and in [START_REF] Frisch | A Cauchy integral equation method for analytic solutions of half-space convolution equations[END_REF] for monochromatic scattering; see also Frisch (2022, p. 53). We thus obtain

λ(ν, ξ) j(ν, ξ) = ω b ξ k(ν, ξ) +∞ -∞ dν ′ j(ν ′ , ξ) ν ′ -ν + ξ p * (ν)t(ν, ξ) + k(ν, ξ) +∞ -∞ p * (ν ′ ) ν ′ -ν dν ′ , (15)
where

k(ν, ξ) = 1/2ν, ν > ξ, 0 0 < ν < ξ. , (16)
t(ν, ξ) ≡ ∞ ξ k(ν ′ , ξ) 1 ν ′ + ν + 1 ν ′ -ν dν, (17)
and

λ(ν, ξ) ≡ 1 -ω b ξ t(ν, ξ). (18)
The symbol means that the integral has to be taken in principal part. Assuming that (ν) is a known function, Eq. ( 15) is for j(ν, ξ) a singular integral equation with a Cauchy kernel. It is similar to Eq. ( 18) in [START_REF] Hemsch | Radiative transfer with partially coherent scattering[END_REF]. We note that a Laplace transform applied to the integral equation for J(τ, x) would lead to a singular integral equation, adjoint to Eq. (15).

To solve this singular integral equations and others of the same type encountered in this paper, we use a standard method of solution introduced by [START_REF] Carleman | Sur la résolution de certaines équations intégrales[END_REF]. It amounts is to transform the integral equation into a boundary value problem in the complex plane, for the Hilbert transform of the unknown function. We recall that the Hilbert transform of a function f (t) is the integral

h(z) ≡ 1 2i π C f (t) t -z dt, (19)
where z belongs to the complex plane and C is a given curve in the complex plane. In the following we use the symbolic notation

h(z) = H C [ f (t)], (20)
L(z, ξ) ≡ 1 -ω b ξT (z, ξ), (22)
where z belongs to the complex plane. T (z, ξ), which is actually the Fourier transform of the kernel K M (τξ), may be written as The function L(z, ξ) is analytic in the complex plane cut along C(ξ) and goes to one at infinity. The Plemelj formulae applied to T (z, ξ) and L(z, ξ), for ν ∈ C(ξ), lead to

248 T (z, ξ) = C(ξ) k(ν, ξ) 1 ν -z dν. ( 23
T + (ν, ξ) -T -(ν, ξ) = 2i π k(ν, ξ), (24)
T + (ν, ξ) + T -(ν, ξ) = 2 t(ν, ξ), (25) L + (ν, ξ) -L -(ν, ξ) = -2i πω b ξ k(ν, ξ), (26) L + (ν, ξ) + L -(ν, ξ) = 2λ(ν, ξ). ( 27
)
An expansion of L(z, ξ) for z → 0 leads to

L(z, ξ) ≃ ǫ b -(1 -ǫ b ) z 2 3ξ 2 , ( 28
)
hence to

L(0, ξ) = ǫ b . ( 29
)
The parameter ǫ b , defined by

ǫ b = 1 -ω b = 1 -(1 -ǫ) b. ( 30
)
is positive and smaller than unity since b ∈ [0, 1].

Other properties of L(z, ξ) are easily derived from those of those of the monochromatic dispersion function corresponding b = ξ = 1. A very important one is that L(z, ξ) has a pair of real zeroes on the real axis in the interval [-ξ, +ξ]. They are given by the zeroes of λ(ν, ξ) at ±ν 0 (ξ) (see Fig. 1). The dispersion function has no other zero. The spectrum of the transport operator has thus two discrete eigenvalues at ±ν 0 (ξ) and a continuous spectrum made of the two infinite half-lines ] -∞, -ξ] and [+ξ, +∞[. The integration over ν in Eq. ( 23) is easily carried out and leads to the explicit expression

L(z, ξ) = 1 -ω b ξ 1 2z ln ξ + z ξ -z , (31)
which shows that L(z, ξ) is actually a function of the complex variable

z = z/ξ. (32)
Moreover, when b is independent of ξ, we can write L(z, ξ) = L(z) with

L(z) = 1 -(1 -ǫ b ) 1 2z ln 1 + z 1 - z . ( 33
)
The zeroes of L(z) are located at ±u 0 , with u 0 solution of L(u 0 ) = 0. Thus when b is a constant, ν 0 (ξ) = ξu 0 . The inverse relation giving the value of ξ for a given position ν of the positive zero

is ξ 0 (ν) = ν/u 0 . When b depends on ξ, we have ξ 0 (ν) = -1 ν 0 (ν), where -1 ν 0 is the inverse function of ν 0 .
The partial derivatives of the dispersion function at the points ±ν 0 (ξ) where it has its zeroes play an important role in the calculation of j(ν, ξ) and (ν). At these points L(z, ξ) = λ(ν, ξ), with ν = ℜ(z). Using Eq. ( 33), one readily obtains

∂λ(ν, ξ) ∂ν | ±ν 0 = ± 1 ξu 0       ǫ b -u 2 0 1 -u 2 0       , (34)
and

∂λ(ν, ξ) ∂ξ | ±ν 0 = ∓ u 0 ν 0       ǫ b -u 2 0 1 -u 2 0       . ( 35
)
The derivatives with respect to ν and ξ have opposite signs, the derivative with respect to ν being negative and the other one positive. Their ratio is -1/u 0 .

Article number, page 4 of 16

Hélène Frisch: Exact solutions for radiative transfer with partial frequency redistribution

The infinite medium

We now apply the Carleman method to solve the singular integral equation for j(ν, ξ) written in Eq. ( 15). This method leads to a boundary value problem in the complex plane for the Hilbert transforms

N ∞ (z, ξ) ≡ H +∞ -∞ [ j(ν, ξ)], N∞ (z) ≡ H +∞ -∞ [ (ν)], (36)
and

N ∞ Q (z) ≡ H +∞ -∞ [q * (ν)], P ∞ (z) ≡ H +∞ -∞ [p * (ν)]. ( 37
)
By construction, these four Hilbert transforms are analytic in the complex plane cut along the real axis. Provided that the integral of Q * (τ) over τ has a finite value, these functions will be well defined and go to zero as z goes to infinity.

In Section 4.1 we determine N ∞ (z, ξ) in terms of N∞ (z) and in Section 4.2 N∞ (z) by an integration over ξ of N ∞ (z, ξ). The functions j(ν, ξ) and (ν) are then given by the jumps of N ∞ (z, ξ)

and N∞ (z) across the real axis (see Eq. ( 21)). Explicit expressions of j(ν, ξ) and (ν) are given in Section 4.3 for

Q * (τ) = δ(τ), that
is for the Green function of the problem.

There is a remark that we wish to make about the definitions of the Hilbert transforms in Eq. ( 36). In this equation, the integration over ν is performed over the full real axis. As mentioned above, λ(ν, ξ) has zeroes in the interval [-ξ, +ξ] at ±ν 0 (ξ), and since the right-hand side in Eq. ( 15) has no reason to be zero at these points, the function j(ν, ξ) will contain Dirac distributions at ±ν 0 (ξ). Hilbert transforms are usually introduced for Hölder continuous functions (e.g., [START_REF] Muskhelishvili | Singular Integral Equations[END_REF]), but they can still be defined for a Dirac distribution [START_REF] Pandey | The Hilbert transform of Schwartz distributions and applications[END_REF] and the Plemelj formulae remain applicable (p. 66 Frisch 2022).

We have checked that the results obtained with Hilbert transforms defined on ] -∞, +∞[ in Eq. ( 36) are identical to those that can be obtained when the Hilbert transforms are defined over

C(ξ) =] -∞, -ξ] ∪ [+ξ, +∞[
. This other approach is carried out in Section 5 devoted to the semi-infinite medium.

4.1. The frequency dependent Hilbert transform N ∞ (z, ξ)

Making use of the expression of L(z, ξ) in Eq. ( 22) and of the Plemelj formulae for T (z, ξ), L(z, ξ), and the Hilbert transforms in Eq. ( 36), we can rewrite Eq. ( 15) as

L + (ν, ξ)N + ∞ (ν, ξ) -L -(ν, ξ)N - ∞ (ν, ξ) = 1 ω b [1 -L + (ν, ξ)]P + ∞ (ν) -[1 -L -(ν, ξ)]P - ∞ (ν) . ( 38
)
The problem is to derive N ∞ (z, ξ) from this equation. We now remark that Eq. ( 38) is equivalent to

F + (ν, ξ) -F -(ν, ξ) = 0, ν ∈] -∞, +∞[, (39)
where

F(z, ξ) ≡ L(z, ξ)N ∞ (z, ξ) - 1 ω b [1 -L(z, ξ)]P ∞ (z). ( 40
)
By construction, F(z, ξ) is analytic in the complex plane cut along the real axis ] -∞, +∞[ and according to Eq. ( 39), the jump of F(z, ξ) across this axis is zero. Hence F(z, ξ) is an entire function, which can be determined by its behavior at infinity, by application of the Liouville theorem (e.g. [START_REF] Carrier | Functions of a Complex Variable[END_REF].

For z → ∞, N ∞ (z, ξ) and P ∞ (z) tend to 0, since they are defined as Hilbert transforms, and L(z, ξ) → 1. Hence, F(z, ξ) → 0 as z → ∞ and by the Liouville theorem F(z, ξ) is zero for all z. We 333 thus obtain

334 L(z, ξ)N ∞ (z, ξ) = 1 ω b [1 -L(z, ξ)]P ∞ (z). ( 41
)
Since L(z, ξ) has zeroes at ±ν 0 (ξ) ∈ [-ξ, +ξ], Eq. ( 41) cannot 335 be straightforwardly divided by L(z, ξ). We must distinguish be-336 tween the regions z ∈ [-ξ, +ξ] and z outside this interval.

337

For z [-ξ, +ξ], we can divide Eq. ( 41) by L(z, ξ). Using 338 also the expression of L(z, ξ) in Eq. ( 22), we obtain

339 N ∞ (z, ξ) = 1 ω b 1 L(z, ξ) -1 P ∞ (z). ( 42
)
In the interval [-ξ, +ξ], according to Eq. ( 16), k(ν, ξ) = 0, 340 hence Eq. ( 15) reduces to

341 λ(ν, ξ) j(ν, ξ) = ξ p * (ν) t(ν, ξ). ( 43
)
Because of the zeroes of λ(ν, ξ) at ±ν 0 (ξ), the solution of Eq. ( 43) 342 has the general form

343 j(ν, ξ) = f (ν, ξ)[δ(ν -ν 0 (ξ)) + δ(ν + ν 0 (ξ))] + p * (ν)]P ξ t(ν, ξ) λ(ν, ξ) , (44)
where P stands for Cauchy principal part and δ(ν ± ν 0 (ξ)) are 344 Dirac distributions at ±ν 0 (ξ). The function f (ν, ξ) can be deter-345 mined by considering the limit z → ±ν 0 (ξ) in Eq. ( 41). In the 346 right-hand side we can set L(z, ξ) = 0. In the left-hand side, the 347 only terms giving a non-zero contribution come from the Dirac 348 distributions in j(ν, ξ). We then Taylor expand L(z, ξ) around its 349 zeroes at ±ν 0 (ξ), that is assume

L(z, ξ) ≃ (z ± ν 0 )∂ ν L(±ν 0 , ξ), 350 where ∂ ν L(±ν 0 , ξ) = ∂ ν λ(±ν 0 , ξ)
is the partial derivative of 351 λ(ν, ξ) with respect to ν taken at ±ν 0 (ξ). Equation ( 41) thus leads 352 to

353 f (±ν 0 , ξ)∂ ν λ(±ν 0 , ξ) = - 2i π ω b P ∞ (±ν 0 ) , (45)
where

354 P ∞ (±ν 0 ) ≡ 1 2 P + ∞ (±ν 0 ) + P - ∞ (±ν 0 ) . (46)
We recall that P ∞ (z) is analytic in the complex plane cut along 355 the real axis. The derivative ∂ ν λ(±ν 0 , ξ) is given in Eq. ( 34).

356

The expressions that we have obtained for N ∞ (z, ξ) and 357 f (±ν 0 , ξ) are still containing the unknown function N∞ (z). We 358 now proceed to the determination of the latter. Knowing how to properly define N ∞ (z, ξ) for all z, we can use 361 Eq. ( 42) to determine N∞ (z). Separating the contributions from 362 q * (ν) and (ν) and integrating this equation over ξ, we obtain 363

Ω(z) N∞ (z) = N ∞ Q (z) ϕ 0 0 W(ξ)ξ T (z, ξ) L(z, ξ) dξ, (47)
where W(ξ) is the weight function defined in Eq. ( 8), and

364

Ω(z) ≡ 1 -ω(1 -b) ϕ 0 0 W(ξ)ξ T (z, ξ) L(z, ξ) dξ. ( 48
)
The function Ω(z) plays for N∞ (z) the role played by the dis-365 persion function L(z, ξ) for N ∞ (z, ξ). It will be referred to as the 366 generalized dispersion function. Equation ( 48) shows that Ω(z) 367

A&A proofs: manuscript no. mixture-aa-general ν continuous spectrum

ξ ϕ 0 0 ϕ 0 ϕ 0 u 0 ϕ 0 u 0 discrete spectrum ν 3 ξ 3 II I III Fig. 2.
Properties of the spectrum of L(ν, ξ) in the variables ν and ξ. The line ν = ξ shows the separation between the continuous and the discrete spectrum. The dotted-dashed line ν = u 0 ξ shows the location of the zeroes of L(z, ξ), with u 0 the positive zero of L(z). The domains I, II, and III corresponds respectively to ν > ϕ 0 , ϕ 0 u 0 < ν < ϕ 0 and 0 < ν < ϕ 0 u 0 . ν 3 is a typical values of ν in the domain III and ξ 3 is the value of ξ such that the positive zero of L(z, ξ) is located at ν 3 . Integrations are along lines parallel to the horizontal ξ-axis for the calculation of Ω ± (ν) and along lines parallel to the vertical ν-axis for the calculations of J(τ, ξ) and J(τ). 42) and ( 45), we obtain

379 N ∞ (z, ξ) = 1 ω b N ∞ Q (z) 1 L(z, ξ) -1 1 Ω(z) , (50)
and

f (±ν 0 , ξ)∂ ν λ(±ν 0 , ξ) = - 2i π ω b NQ (±ν 0 ) 1 Ω(±ν 0 ) . ( 51
)
The expressions that are given here for N∞ (z), N ∞ (z, ξ), and f (±ν 0 , ξ) have been obtained with the assumption that b is a con- 4.3. Explicit expressions of j(ν, ξ) and (ν)

Explicit expressions of j(ν, ξ) and (ν) are given below for the Green function. They are obtained by setting

Q * (τ) = δ(τ). (52)
The function J(τ, ξ) is the regular part of the Green function, usually referred to as the resolvent function. The primary source term in the integral equation for J(τ, ξ) (see Eq. ( 10)) is now ξK M (|τ|ξ), the inverse Laplace transform of which is

ξ k(ν, ξ) = - 1 2i πω b [L + (ν, ξ) -L -(ν, ξ)]. (53)
This term must be added to the integral equation for j(ν, ξ) given in Eq. ( 15), while simultaneously setting p * (ν

) = ω(1 -b) (ν).
Equation ( 41) for the frequency dependent Hilbert transform

N ∞ (z, ξ) becomes L(z, ξ)N ∞ (z, ξ) = [1 -L(z, ξ)] 1 -b b N∞ (z) + 1 ω b2i π . ( 54
)
Comparing it with Eq. ( 41), we see that N ∞ Q (z) has been replaced by 1/2i π. It suffices to carry out this change in Eqs. ( 49), ( 50) and ( 51) to obtain N∞ (z), N ∞ (z, ξ) and f (±ν 0 ).

The function (ν) is given by the jump of N∞ (z) across the real axis. Taking advantage of the fact that Ω + (ν) and Ω -(ν) are complex conjugate, we can write

(ν) = 1 ωπ(1 -b) ℑ[-Ω + (ν)] [ℜ[Ω + (ν)]] 2 + [ℑ[Ω + (ν)]] 2 .
(55)

The complete frequency redistribution limit is obtained by setting b = 0 and replacing Ω(z) by the complete frequency redistribution dispersion function L R (z).

The case of j(ν, ξ) is slightly more complex since we have to distinguish between the discrete part of the spectrum, which contains the zeroes of L(z, ξ) at ±ν 0 (ξ), and the continuous part for ν ∈ C(ξ). For ν ∈ [-ξ, +ξ], j(ν, ξ) contains two Dirac distributions (see Eq. ( 44)). Equations ( 50) and (51) lead to

j d (ν, ξ) = 1 ωπ b ℑ[-Ω + (ν)] Ω + (ν)Ω -(ν) P 1 -λ(ν, ξ) λ(ν, ξ) - 1 ω b 1 ∂λ ν (ν, ξ) 1 Ω(ν) [δ(ν -ν 0 (ξ)) + δ(ν + ν 0 (ξ))]. ( 56
)
The subscript d stands for discrete. The angle bracket is defined in Eq. ( 46). One can easily check that the right-hand side is positive.

For ν ∈ C(ξ), j(ν, ξ) is given by the jump of N ∞ (z, ξ). Equation (50) and the Plemelj formulae for L(z, ξ) lead to

j c (ν, ξ) = ξ k(ν, ξ) L + (ν, ξ)L -(ν, ξ) + 1 2i πω b (1 -Ω(z))(1 -L(z, ξ)) Ω(z)L(z, ξ) j , (57)
where the subscript j stands for jump across the real axis and the subscript c for continuous.

In the monochromatic limit Ω(z) = 1 (see e.g. Eq. ( 48)). The right-hand side of Eq. ( 56) is thus reduced to its distributional term and the right-hand side of Eq. ( 57) to its first term.

Article number, page 6 of 16 Hélène Frisch: Exact solutions for radiative transfer with partial frequency redistribution 4.4. The source function in the physical space Having determined j(ν, ξ) and (ν), we can now calculate the source function

S (τ, ξ) = ω[b J(τ, ξ) + (1 -b) J(τ)] + Q * (τ). ( 58
)
Without loss of generality, we can consider only positive values of τ. Then we have

J(τ, ξ) = ∞ 0 j(ν, ξ) e -ντ dν, J(τ) = ∞ 0 (ν) e -ντ dν. ( 59
)
We recall that ξ = ϕ(x), with ϕ(x) the absorption line profile. The integrations over ν in Eq. ( 59) amount to an integration along the vertical ν-axis across the three domains shown in Fig. 2. Explicit expressions of Ω ± (ν) for each domains are given in Appendix A.1. No special precautions are needed, but it is worth discussing the role of the zero of L(z, ξ) for ν in domain III,

where the imaginary part of Ω + (ν) given in Eq. (A.12) contains a term of the form

-πω(1 -b) W(ξ 0 )ξ 0 t(ν, ξ 0 ) ∂ ξ L(ν, ξ 0 ) . ( 60
)
We recall that ξ 0 is a function of ν, given by ξ 0 = u 0 /ν when b is a constant (see Section 3). The contribution of the zero at ξ 0 (ν) appears thus through the derivative of the dispersion function, but has a different effect on J(τ, ξ) and J(τ).

For ν in domain III, j(ν, ξ) has a distributional term f (ν 0 (ξ), ξ)δ(νν 0 (ξ)) (see Eq. ( 56)). After an integration over ν, weighted by exp(-ντ), the function J(τ, ξ) will have a term of the form exp[-ν 0 (ξ)τ], with a coefficient depending on the derivative of the dispersion function at ν 0 (ξ). For the monochromatic Green function, it is this term which will determine its large τ asymptotic behavior (see e.g., Ivanov 1973, p. 117).

For J(τ), the partial derivative ∂ ξ L(ν, ξ 0 ) will come from the values of ν in domain III. One can interpret the result by remarking that the zeroes of L(z, ξ) are embedded in the singular line of N∞ (z). In the limit of complete frequency redistribution, the contribution of the zeroes disappears since Ω(z) becomes equal to L R (z), which is free of zeroes. In this limit, the Green function decreases algebraically at infinity (see e.g., Ivanov 1973, p. 159).

For a combination of monochromatic scattering and complete frequency redistribution, the exponential behavior is hidden by the large τ algebraic behavior of the complete frequency redistribution contribution. This point is further discussed in Section 6.

The semi-infinite medium

We now consider a semi-infinite plane-parallel medium, where the space variable τ varies from 0, at the surface, to ∞ in the interior. We also assume that there is no incident radiation field at τ = 0. An incident field can always be transformed into an interior source by separating the contribution of the incident field from that of the diffuse field created by the scattering. The functions J(τ, ξ) and J(τ) for τ > 0 are defined as shown in Eqs. ( 12) to (13). Now ν has only positive values.

We first determine j(ν, ξ) (Section 5.1) and then (ν) by integration over ξ (Sections 5.2 and 5.3). Exact solutions of half-space problems always depend on an half-space auxiliary function derived from the dispersion function, a property well established since [START_REF] Wiener | Über eine Klasse singulärer Integralgleichungen, Sitzungsberichte der Preussichen Akademie[END_REF]. The same situation holds here. We will introduce two half-space auxiliary functions, namely X(z, ξ) constructed with L(z, ξ) and X(z), constructed with Ω(z). We use a standard construction method based on the solution of a homogeneous Riemann-Hilbert problem 474 (e.g., [START_REF] Ablowitz | Complex Variables, Introduction and Applications[END_REF]. In Sections 5.1, 5.2, and 5.3, the 475 branching ratio b is a constant. In Section 5.4, we indicate the 476 main changes that occur when b becomes ξ-dependent. Finally 477 we show in Section 5.5 how to calculate the emergent intensity. 478 5.1. The frequency dependent function j(ν, ξ)

479

The singular integral equation for j(ν, ξ) is given by Eq. ( 15), 480 with the domain of definition of ν restricted to [0, ∞[. This means 481 that we now have to solve

482 λ(ν, ξ) j(ν, ξ) = ω b ξ k(ν, ξ) ∞ 0 j(ν ′ , ξ) dν ′ ν ′ -ν + ξ p * (ν) t(ν, ξ) + k(ν, ξ) ∞ 0 p * (ν ′ ) dν ′ ν ′ -ν , (61)
where p * (ν) has been defined in Eq. ( 14). For the full-space case, 483 we have introduced N ∞ (z, ξ), the Hilbert transform of j(ν, ξ) over 484 the full real axis. Here, we consider separately the interval ν ∈ 485 [0, ξ], where j(ν, ξ) has a distributional solution, and the interval 486 ν ∈ [ξ, ∞[ where we explain why an auxiliary function X(z, ξ) is 487 needed to solve the integral equation for j(ν, ξ). In this interval, k(ν, ξ) is identically zero (see Eq. ( 16)) and 490 λ(ν, ξ) = 0 at ν = ν 0 (ξ). The solution of Eq. ( 61) has thus the 491 same form as in the full-space case, namely

492 j(ν, ξ) = ξ p * (ν) P t(ν, ξ) λ(ν, ξ) + f (ν, ξ)δ(ν -ν 0 (ξ)), (62)
λ(ν, ξ) j(ν, ξ) = ω b ξ k(ν, ξ) ∞ ξ j(ν ′ , ξ) dν ′ ν ′ -ν + ω b ξ k(ν, ξ) f (ν 0 , ξ) ν 0 -ν + G(p * , ν, ξ). (63)
In G(p * , ν, ξ) we have regrouped the terms depending on p * (ν). 500 After some simple algebra, which makes use in particular of the 501 definition of λ(ν, ξ) given in Eq. ( 18), we can write 502

G(p * , ν, ξ) = ξk(ν, ξ) ξ 0 p * (ν ′ ) λ(ν ′ , ξ) dν ′ ν ′ -ν + ξ p * (ν)t(ν, ξ) + k(ν, ξ) ∞ ξ p * (ν ′ ) dν ′ ν ′ -ν . ( 64
)
The principal part in the integration over the interval [0, ξ] is 503 due to the zero of λ(ν, ξ) at ν = ν 0 (ξ). In the monochromatic 504 scattering limit, G(p * , ν, ξ) depends only on the primary source 505 term since p * (ν) = q * (ν).

506

To solve Eq. ( 63) for j(ν, ξ), we first transform it into a 507 boundary value problem in the complex plane. We introduce the 508 Hilbert transforms

509 N c (z, ξ) ≡ H ∞ ξ [ j(ν, ξ)], P c (z, ξ) ≡ H ∞ ξ [p * (ν)]. ( 65
)
Article number, page 7 of 16

A&A proofs: manuscript no. mixture-aa-general

The subscript c stands for continuous. In P c (z, ξ) the dependence on ξ comes from the integration domain. The two functions in Eq. ( 65) are analytic in the complex plane cut along [ξ, ∞[ and go to zero at infinity.

Introducing the Plemelj formulas for L(z, ξ), N c (z, ξ), and P c (z, ξ) into Eq. ( 63), we obtain after some elementary algebra

L + (ν, ξ)M + (ν, ξ) -L -(ν, ξ)M -(ν, ξ) = 1 ω b m(ν, ξ), (66)
with

m(ν, ξ) = p * (ν) -L + (ν, ξ) -L -(ν, ξ) 1 2i π ξ 0 p * (ν ′ ) λ(ν ′ , ξ) dν ′ ν ′ -ν . ( 67
)
Here M ± (ν, ξ) are the limiting values above and below the real axis of the function M(z, ξ) defined by

M(z, ξ) ≡ N c (z, ξ) + 1 2i π f (ν 0 , ξ) ν 0 -z + 1 ω b P c (z, ξ). ( 68
)
The functions M(z, ξ) and L(z, ξ) have different analyticity prop-

erties : M(z, ξ) is analytic in the complex plane cut along [ξ, ∞[, while the singular cut of L(z, ξ) is along ] -∞, -ξ] ∪ [+ξ, +∞[.
A powerful method for solving such equations is to introduce an auxiliary function X(z, ξ), with the same analyticity domain as the unknown function, here M(z, ξ) (see e.g., [START_REF] Ablowitz | Complex Variables, Introduction and Applications[END_REF]. This means that we have to construct a function X(z, ξ)

analytic in the complex plane cut along [ξ, ∞[. In addition it should be free of singularities and solution of the homogeneous Riemann-Hilbert problem

X + (ν, ξ) X -(ν, ξ) = L + (ν, ξ) L -(ν, ξ) , ν ∈ [ξ, ∞[. (69)
The standard method to solve such an equation is to take its logarithm and then use the Plemelj formulae. Imposing the condition that X(z, ξ) is free of singularity, one finds the exact expression

X(z, ξ) = 1 1 -z exp 1 π ∞ ξ θ(ν, ξ) dν ν -z , ( 70
)
where θ(ν, ξ) is the argument of L + (ν, ξ) (e.g., [START_REF] Muskhelishvili | Singular Integral Equations[END_REF][START_REF] Gakhov | Boundary Value Problems[END_REF][START_REF] Ablowitz | Complex Variables, Introduction and Applications[END_REF]Frisch 2022, p. 82). In the following, we shall make use of the function

X * (z, ξ) ≡ (ν 0 -z)X(z, ξ). ( 71
)
It satisfies the factorization relation

X * (z, ξ)X * (-z, ξ) = L(z, ξ), (72)
which, combined with

L(0, ξ) = ǫ b = 1 -ω b, leads to X * (0, ξ) = √ ǫ b . In the monochromatic limit, i.e., when b = ξ = 1, X * (z, ξ)
is simply related to the monochromatic Chandrasekhar's Hfunction (see Section 5.5).

To pursue the determination M(z, ξ), we multiply Eq. ( 66) by (ν 0ν). Making use of Eq. ( 69), we obtain

[X * ] + (ν, ξ)M + (ν, ξ) -[X * ] -(ν, ξ)M -(ν, ξ) = [X * ] + (ν, ξ) L + (ν, ξ) m(ν, ξ) ω b . ( 73
)
The

M(z, ξ) ≡ H ∞ ξ [X * ] + (ν, ξ) L + (ν, ξ) m(ν, ξ) . ( 74
)
This function goes to zero at infinity and is free of singularities. 546 Proceeding as in Section 4, we introduce the function 547

F(z, ξ) ≡ (ν 0 -z)X(z, ξ)M(z, ξ) - 1 ω b M(z, ξ). ( 75
)
By construction F(z, ξ) is analytic in the complex plane cut along 548 the real semi-infinite line [ξ, ∞[. According to Eq. ( 73) its jump 549 across [ξ, ∞[ is zero. It is also free of singularity, since the sin-550 gularity in M(z, ξ) due to the term f (ν 0 , ξ)/(ν 0z) (see Eq. ( 68)) 551 is canceled by the factor (ν 0z). One may thus conclude that 552 F(z, ξ) is an entire function, which can be determined by its be-553 havior at infinity. Remembering that X(z, ξ) ≃ -1/z as z → ∞, 554 we see that all the terms in Eq. ( 75) go to zero at infinity. Hence 555

F(z, ξ) = 0, ( 76
)
for all values of z and ξ. We thus obtain

556 N c (z, ξ) = 1 ω b M(z, ξ) (ν 0 -z)X(z, ξ) - 1 2i π f (ν 0 , ξ) ν 0 -z - 1 ω b P c (z, ξ). ( 77
)
Since N c (z, ξ) has been defined as a Hilbert transform, it should 557 be free of singularity in its analyticity domain. Equation ( 77) 558 shows that N c (z, ξ) has a singularity at z = ν 0 . It is readily elimi-559 nated by choosing

560 f (ν 0 , ξ) = 2i π ω b M(ν 0 , ξ) X(ν 0 , ξ) . ( 78
)
The expressions of N c (z, ξ) and f (ν 0 , ξ) given above still depend 561 on (ν). Before performing any integration over ξ, we need the 562 Hilbert transform of j(ν, ξ) defined over the full interval [0, ∞[. 563 Summing the contributions from Eqs. ( 62) and ( 77), we obtain 564

1 2i π ∞ 0 j(ν, ξ) ν -z dν = 1 ω b M(z, ξ) X * (z, ξ) - 1 2i π 1 ω b ∞ ξ p * (ν) dν ν -z -ξ ξ 0 p * (ν) t(ν, ξ) λ(ν, ξ) dν ν -z . (79) It is shown in Appendix B that 565 M(z, ξ) = -X * (z, ξ) 1 2i π ξ 0 p * (ν) λ(ν, ξ) dν ν -z + 1 2i π ∞ 0 p * (ν) X * (-ν, ξ) dν ν -z . ( 80
)
Making use of the relation λ(ν, ξ) = 1ω bξ t(ν, ξ), we obtain

566 1 2i π ∞ 0 j(ν, ξ) ν -z dν = - 1 ω b 1 2i π ∞ 0 p * (ν) dν ν -z + 1 ω b 1 X * (z, ξ) 1 2i π ∞ 0 p * (ν) X * (-ν, ξ) dν ν -z , (81)
In the monochromatic limit, the only terms remaining in the 567 right-hand side of Eq. ( 81) are those containing q * (ν). Equa-568 tion (81) can also be obtained by working directly with the 569 Hilbert transforms of j(ν, ξ) and (ν) defined over [0, ∞[ as done 570 for the full-space case and as in [START_REF] Hemsch | Radiative transfer with partially coherent scattering[END_REF]. 571

The introduction of the half-space auxiliary function X(z, ξ) ap-572 pears more naturally when the two intervals [0, ξ] and [ξ, ∞[ are 573 treated separately. When Q * (τ) = δ(τ), the contribution from 574 the primary source in the right hand-side of Eq. ( 81) is given by 575 [[X * (z, ξ)] -1 -1]/(2i πω).

576

Article number, page 8 of 16

Hélène Frisch: Exact solutions for radiative transfer with partial frequency redistribution 5.2. The frequency integrated function (ν)

Proceeding as in the full-space case we integrate Eq. ( 81) over ξ. Whereas for the full-space case, it was possible to obtain an explicit expression of (ν) by way of its Hilbert transform (see Section 4.2), for the half-space case only a Fredholm integral equation can be obtained for (ν).

Making us of the factorization relation in Eq. ( 72) and introducing the Hilbert transforms

N(z) = H ∞ 0 [ (ν)],
and

N Q (z) = H ∞ 0 [q * (ν)], (82)
we can write Eq. ( 81) as

1 b N(z) = - 1 ω b N Q (z) + 1 ω b 1 2i π ϕ 0 0 W(ξ) L(z, ξ) ∞ 0 X * (-z, ξ) X * (-ν, ξ) p * (ν) dν ν -z dξ. ( 83
)
As pointed out in [START_REF] Hemsch | Radiative transfer with partially coherent scattering[END_REF], the singular integral in the right-hand side of Eq. ( 83) has a Cauchy kernel 1/(νz) multiplied by 1/X * (-ν, ξ). Singular integral equations with Cauchy-type kernels are treated in Muskhelishvili (1953, Chapter 6,). We now follow the method described in this reference and also applied in [START_REF] Hemsch | Radiative transfer with partially coherent scattering[END_REF]. It is built on the separation of the Cauchy-type kernel into a "dominant" part and a "Fredholm" part. This amounts to introduce a function r(z, ν, ξ) defined by

X * (-z, ξ) X * (-ν, ξ) 1 ν -z = 1 ν -z + r(z, ν, ξ). (84)
Regrouping the integrals with the Cauchy kernels and writing Ω(z), defined in Eq. ( 48), as

Ω(z) ≡ 1 - 1 -b b ϕ 0 0 W(ξ) 1 L(z, ξ) -1 dξ, (85)
we can rewrite Eq. ( 83) as

Ω(z)[ N(z) + 1 ω(1 -b) N Q (z)] = G(z) + R(z), (86)
where

R(z) ≡ 1 -b b ∞ 0 (ν)V(z, ν) dν, (87)
G(z) ≡ 1 ω(1 -b) N Q (z) + 1 ω b ∞ 0 q * (ν)V(z, ν) dν, (88)
and

V(z, ν) ≡ 1 2i π ϕ 0 0 W(ξ) L(z, ξ) r(z, ν, ξ) dξ. (89)
It is the term R(z) in Eq. ( 86), which prevents the construction of an exact solution for (ν). This term, which comes from the Fredholm part of the kernel, does not exist in the full-space case.

A Fredholm integral equation for (ν)

It is not possible to derive N(z) directly from Eq. ( 86) because N(z) is analytic in the complex plane cut along [0, ∞[, while the domains of analyticity of Ω(z), R(z) and G(z) are the complex plane cut along ] -∞, +∞[. However, for ν ∈ [0, ∞[, we can 608 derive from this equation a boundary value problem, namely

609 Ω + (ν)[ N+ (ν) + 1 ω(1 -b) N + Q (ν)] -Ω -(ν)[ N-(ν) + 1 ω(1 -b) N - Q (ν)] = ρ(ν) + g(ν), (90)
with

610 ρ(ν) ≡ R + (ν) -R -(ν), g(ν) ≡ G + (ν) -G -(ν). (91)
Equation ( 87) shows that ρ(ν) may be written as

611 ρ(ν) = 1 -b b ∞ 0 (ν ′ )V(ν, ν ′ ) dν ′ , (92)
where V(ν, ν ′ ) is the jump of the function V(z, ν) defined in 612 Eq. ( 89). This jump is calculated in Appendix B (see Eqs. (B.12), 613 (B.13), and (B.14)). The jump g(ν) may be written as

614 g(ν) = 1 ω(1 -b) q * (ν) + 1 ω b ∞ 0 q * (ν ′ )V(ν, ν ′ ) dν ′ . (93)
For complete frequency redistribution r(z, ν, ξ) is zero, hence 615 ρ(ν) = 0 and g(ν) is reduced to its first term.

616

Because the analyticity domain of Ω(z) is cut along the real 617 axis, the solution of Eq. ( 90) requires the introduction of an aux-618 iliary function, denoted X(z), analytic in the complex plane cut 619 along [0, ∞[, free of zero and singularity, and such that 620

X+ (ν) X-(ν) = Ω + (ν) Ω -(ν) , ν ∈ [0, ∞[. (94)
As shown in Section A.1, Ω(z) is free of zeroes. Hence X(z) is 621 given by 622

X(z) ≡ exp 1 π ∞ 0 θ(ν) dν ν -z , (95)
where θ(ν) is the argument of Ω + (ν). The definition of X(z) 623 shows that X(z) → 1 as z → ∞. The function X(z) satisfies the 624 factorization relation

625 X(-z) X(z) = Ω(z), (96)
for z outside the cut [0, ∞[. In the limit b = 0, Ω(z) becomes 626 identical to the dispersion function for complete frequency redis-627 tribution L R (z) (see Section A.2) and the function X(z) identical 628 to the X-function for complete frequency redistribution, hence-629 forth denoted X R (z).

630

Knowing that Ω(z) is free of zeroes, we can divide Eq. ( 90) 631 by Ω + (ν) and make use of Eq. ( 94). We also introduce the two 632 Hilbert transforms

633 G(z) ≡ H ∞ 0 X+ (ν) Ω + (ν) g(ν) = H ∞ 0 g(ν) X(-ν) , ( 97
) 634 R(z) ≡ H ∞ 0 X+ (ν) Ω + (ν) ρ(ν) = H ∞ 0 ρ(ν) X(-ν) . ( 98
)
They are analytic in the complex plane cut along [0, ∞[. We can 635 now write Eq. ( 90) as

636 X+ (ν)       N+ (ν) + N + Q (ν) ω(1 -b)       -X-(ν)       N-(ν) + N - Q (ν) ω(1 -b)       = [G + (ν) -G -(ν)] + [R + (ν) -R -(ν)]. ( 99
)
Article number, page 9 of 16 A&A proofs: manuscript no. mixture-aa-general

Following the method already used several times, we introduce

(ν) - ∞ 0 (ν ′ )κ(ν, ν ′ ) dν ′ = - 1 ω(1 -b) q * (ν) + η(ν). (102)
The kernel κ(ν, ν ′ ) is given by complete frequency redistribution in a semi-infinite medium (see 666 e.g., [START_REF] Ivanov | Transfer of Radiation in Spectral Lines, Washington Governement Printing Office[END_REF]Frisch 2022, p. 104). In all the preceding sections it is assumed that b, the branching ratio between coherent and incoherent scattering is a constant. We indicate in this section the main changes that have to be implemented on the results obtained for a semi-infinite medium when dealing with a frequency dependent branching ratio b(ξ), such that 0 ≤ b(ξ) ≤ 1.

We can right away say that the main results are not modified : existence of an exact solution for the full-space case and of a Fredholm integral equation for the half-space one. The reason is that the dispersion function L(z, ξ), the generalized dispersion function Ω(z), and the half-space auxiliary functions X(z, ξ) and X(z) keep the same structural properties with respect to analyticity, to zeroes, and to their behavior at infinity. Of course, all the results which involve an integration over ξ will have a different expression.

For L(z, ξ), the two zeroes at ±ν 0 (ξ) are still located in the interval [-ξ, +ξ] , but the simple relation ν 0 (ξ) = ξu 0 , with u 0 the positive zero of L(z) does not hold anymore since L(z, ξ) remains ξ-dependent, even after the introduction of the variable z = z/ξ (see Eq. ( 32)).

For Ω(z), the definition given in Eq. ( 48) becomes

Ω(z) = 1 -ω ϕ 0 0 a(ξ)W(ξ) ξ T (z, ξ) L(z, ξ) dξ, (109)
where a(ξ) = 1b(ξ). The value of Ω(z) for z = 0 given in Eq. (A.2) becomes

Ω(0) = ϕ 0 0 W(ξ) ǫ ǫ b (ξ) dξ, (110)
where ǫ b (ξ) = 1ω b(ξ). Since ν 0 (ξ) is not a linear function of ξ, the dotted-dashed line in Fig. 2 The modifications that have to be brought to L(z, ξ) and Ω(z) will also affect the half-space auxiliary functions X(z, ξ) and X(z), but indirectly only, since their definitions, given in Eqs. ( 69) and (94), are independent of b(ξ).

The starting point for establishing expressions of (ν) and N(z), valid for a ξ-dependent branching ratio, is Eq. ( 81). Proceeding exactly as in Section 5.2, the integration over ξ now yields

Ω(z) N(z) = R b (z) + G b (z), (111)
where

R b (z) = 1 2i π ∞ 0 (ν) ϕ 0 0 a(ξ) b(ξ) W(ξ) L(z, ξ) r(z, ν, ξ) dξ dν, (112)
and

G b (z) = N Q (z) ϕ 0 0 W(ξ) ξ T (z, ξ) L(z, ξ) dξ + 1 2i π 1 1 -ǫ ϕ 0 0 W(ξ) b(ξ)L(z, ξ) ∞ 0 r(z, ν, ξ) q * (ν) dν dξ. (113)
The treatment of Eq. ( 111) goes then exactly as described in Sections 5.2 and 5.3, keeping in mind that b now depends on ξ. For example, the jump 1972). This article shows in particular how the line profile is af-720 fected by a monochromatic component acting at large frequencies. Their results are discussed in some detail in Section 6.

ρ b (ν) = R + b (ν) -R - b (ν) takes the form ρ b (ν) = ∞ 0 V b (ν, ν ′ ) (ν ′ ) dν ′ , (114)
N S (z, ξ) = 1 X * (z, ξ) 1 2i π ∞ 0 ω(1 -b(ξ)) (ν) + q * (ν) X * (-ν, ξ) dν ν -z , (121)
where (ν) is the solution of the Fredholm integral equation

747

For monochromatic scattering, b = 1, and ξ has a constant value, which we take equal to one, following a usual practice. Equation ( 121) straightforwardly leads to

N S (z, ξ) = 1 X * M (z) 1 2i π ∞ 0 1 X * M (-ν) q * (ν) dν ν -z , (122)
where X * M (z) = X * (z, 1) = (ν 0 (1)z)X(z, 1).

For complete frequency redistribution, b = 0. Equation (117) leads to

N S (z, ξ) = ω N(z) + N Q (z). ( 123
)
The expression of N(z) for complete frequency redistribution can be derived from Eq. ( 101) by setting R(z) to zero and g(ν) = q * (ν)/(ω(1b)) in the definition of G(z). We readily obtain

N S (z, ξ) = 1 X R (z) 1 2i π ∞ 0 1 X R (-ν) q * (ν) dν ν -z . ( 124
)
We thus obtain the same expression as for monochromatic scattering, with X * (z), replaced by X R (z). For monochromatic scattering and complete frequency redistribution, N S (z, ξ) depends only on z, a result which is consistent with the fact that the source function depends only on τ.

In these two limits, the emergent intensity may thus be written as

I(0, ξ, µ) = H( µ ξ ) ∞ 0 ξ ξ + µν q * (ν)H( 1 ν ) dν, µ ≥ 0, (125)
with ξ = ϕ(x) for complete frequency redistribution and ξ = 1 for monochromatic scattering. The H-function is defined by

H(z) = 1/X R (-1/z), complete redistribution 1/X * M (-1/z), monochromatic scattering. (126)
For monochromatic scattering, H(z) is the Chandrasekhar Hfunction for isotropic scattering [START_REF] Chandrasekhar | Radiative Transfer[END_REF]. For complete frequency redistribution, the properties of the H-function are discussed in detail in, e.g., [START_REF] Ivanov | Transfer of Radiation in Spectral Lines, Washington Governement Printing Office[END_REF] (see also [START_REF] Frisch | Radiative Transfer, Introduction to exact and asymptotic methods[END_REF]. They are similar to those of the monochromatic Hfunction. For example, they both increase monotonically from H(0) = 1 to 1/ √ ǫ at infinity. The value at infinity is easily derived from the factorization relations for X R (z) and X * M (z) and from L M (0) = L R (0) = ǫ. Equation ( 125) is valid for all positive values of µ, but only those belonging to the interval [0, 1] are needed for the calculation of the emergent intensity. This equation provides a very convenient expression for the emergent intensity, especially when q * (ν) has a simple form. For example, when the primary source has a uniform value Q * , its inverse Laplace transform is q * (ν) = Q * δ(ν). In the monochromatic and complete frequency redistribution limits, the emergent intensity is thus simply given by

I(0, x, µ) = Q * √ ǫ H( µ ϕ(x) ), (127)
with ϕ(x) = 1 in the monochromatic case.

Everywhere in this article it has been assumed implicitly that ǫ 0. The results that have been obtained hold also for ǫ = 0. Indeed, provided b 1, the parameter ǫ b (ξ) will remain strictly positive, smaller than one and the dispersion function L(z, ξ) will have two zeroes in the interval [-ξ, +ξ]. Only in the monochromatic limit b = 1 will there be a difference, since L(z, ξ) will have a double zero at the origin. The method, which is summarized above, shows that a sig-821 nificant effort is needed to find exact solutions for a partial fre- finite medium decreases at infinity as exp(-ν 0 |τ|), where ν 0 is the positive root of the frequency dependent dispersion function L(z, ξ). In contrast, for complete frequency redistribution, the second order moment of the kernel K R (τ) (see its definition in Appendix A.2) is infinite, the dispersion function L R (z) has no zero, and the large scale behavior of the radiation field has the structure of a Lévy walk. The infinite medium Green function decreases algebraically at infinity as a power law, with an exponent depending on the wing behavior of the absorption profile (see e.g., Frisch 2022, Chapters 21, 22). For a linear combination of the two scattering mechanisms, the law of decrease of the Green function varies between the complete frequency redistribution limit for b = 0 and the monochromatic one for b = 1. The frequency dependent term J(τ, ξ) contains an exponential term coming from the distributional term in Eq. ( 56) and also a non exponential one, coming from the other term. Both terms involve partial derivative of the dispersion function taken at ±ν 0 . The frequency averaged term J(τ) has no exponential term (see Eq. ( 55), but the zeroes at ±ν 0 leave their mark through the partial derivative of L(z, ξ) at ±ν 0 . Equations ( 56) and ( 55) show also that the large τ asymptotic behavior of the Green function and more generally the coupling between the two scattering mechanisms is controlled by the generalized dispersion function Ω(z) defined in Eq. ( 48). It has no zero, is identical to the complete frequency redistribution dispersion function L R (z) for b = 0 and has the constant value 1 for b = 1.

Whether the transition between the complete frequency redistribution and the monochromatic regime is rapid or slow and for which values of b it occurs is an interesting question. The theoretical results given above suggests that the complete frequency redistribution regime will be dominant unless b is very close to one since of the three terms in j(ν, ξ) and (ν) only one of them will be responsible for an exponential behavior. This conjecture is in agreement with a large τ asymptotic analysis of the source function S (τ, ξ), which shows that the large τ behavior is indeed dominated by complete frequency redistribution (see e.g., Frisch 2022, p. 569). It seems to be also in agreement with the numerical investigation of the emergent intensity reported in [START_REF] Hemsch | Linear transport theory with partly incoherent scattering[END_REF] and [START_REF] Hemsch | Radiative transfer with partially coherent scattering[END_REF]. Line profiles of the emergent intensity for a semi-infinite medium with a uniform primary source are calculated for a frequency dependent parameter b equal to 0 in the line center and equal to some constant value b W in the wings. Numerical results for b W = 0.2, 0.5, 0.9 show that b W has to be close to 1 for departures from the complete frequency redistribution limit to become significant.

A systematic investigation of the emergent intensity profile or of the Green function in an infinite medium, keeping b independent of the frequency, could perhaps lead to the determination of some critical value of b marking the transition from a pure complete frequency redistribution profile for b = 0 to a pure monochromatic one for b = 1. This investigation can be carried out with the expressions given here or with a direct numerical solution of the radiative transfer equation.

168

  equation into an integral equation is to insert the formal solution 169 of the radiative transfer equation into the definition of the source 170 function. For an infinite medium, this technique leads to 171

Fig. 1 .

 1 Fig. 1. The function λ(ν, ξ), real part of L ± (ν, ξ), as a function of ν, for a given value of the frequency variable ξ = ϕ(x). At the origin λ(0, ξ) = ǫ b = 1ω b and at infinity λ(±∞) = 1. The zeroes at ±ν o (ξ) are also the zeroes of the dispersion function L(z, ξ). The latter is analytic in the complex plane cut along the semi-infinite intervals ℜ(z) = ν ∈] -∞, -ξ] ∪ [+ξ, +∞[ indicated here by two thick lines.where the subscript will indicate the integration curve. The func-223

)

  The notation C(ξ) stands for ] -∞, -ξ] ∪ [+ξ, +∞[, that is the union of the two intervals ] -∞, -ξ] and [+ξ, +∞[. A sketch of 250 λ(ν, ξ) for ν ∈] -∞, +∞[ is shown in Fig. 1. The function T (z, ξ) 251 is obviously an Hilbert transform.

  The frequency integrated Hilbert transform N∞ (z) 360

  is analytic in C | /IR 2 . It is shown in Appendix A.1 that Ω(z) has no 368 zero and in Appendix A.2 that it goes to the complete frequency 369 dispersion function, henceforth denoted L R (z), for b = 0. The 370 subscript R will henceforth stand for complete frequency redis-371 tribution. For b = 1, that is in the monochromatic limit, one 372 simply has Ω(z) = 1. Other properties of Ω(z) are described in f (±ν 0 , ξ). Inserting Eq. (49) into Eqs. (

  382 stant. Changes that are needed when b depends on ξ are easy 383 to make. They are indicated in Section 5.4 for the semi-infinite 384 medium.

  left-hand side of this equation is the jump across [ξ, ∞[ of the function (ν 0z)X(z, ξ)M(z, ξ), which is analytic in the complex plane cut along [ξ, ∞[. The right-hand side can be given a similar structure by introducing the Hilbert transform

  Frequency dependent branching ratio b(ξ)

  will become a function of ξ, increasing between ν 0 (0) = 0 and ν 0 (ϕ 0 ) and remaining below the line ν = ξ. The expressions of Ω ± (ν) given in Appendix A.1 remain valid, provided b and (1b) are replaced by b(ξ) and (1b(ξ)) and included inside the integrals over ξ, whenever necessary. In Eq. (A.12), (1b) should be replaced by (1b(ξ 0 (ν

  Article number, page 10 of 16 Hélène Frisch: Exact solutions for radiative transfer with partial frequency redistribution where V b (ν, ν ′ ) is given by Eq. (B.14) in which the factor (1b)/b is included inside the integral. In Eq. (B.13), which provides 714 the contribution of the zero of L(z, ξ) to V b (ν, ν ′ ), b should be 715 replaced by b(ξ 0 (ν)). for a uniform primary source is stud-718 ied in detail, analytically and numerically, in Hemsch & Ferziger719

  743in Eq. (102). There is thus no explicit expression for N S (z, ξ) 744 and for the emergent intensity, except in the limiting cases of 745 monochromatic scattering and complete frequency redistribu-746 tion.

Fig. 3 .

 3 Fig. 3. Contour in the complex plane for the calculation of the emergent intensity. The function N S (z) is analytic in the complex plane cut along the semi-infinite real line [0, ∞[.6. Summary of the solution method and concluding

  820

822

  quency redistribution problem, which is after all rather elemen-823 tary. A reward for this effort is the possibility to grasp struc-824 tural properties of the problem, such as the coupling between 825 two scattering mechanisms of rather different nature, in a way 826 that no direct numerical simulation can offer. 827 For monochromatic scattering, the kernel K M (τ) has a finite 828 second order moment, a consequence being that its dispersion 829 function has isolated zeroes. This implies that the large scale 830 behaviour of the radiative transfer equation is a diffusion equa-831 tion and also that the monochromatic Green function in an in-832

  The absorp-126 tion profile ϕ(x) is a positive and even function of x, which goes 127 to zero at infinity. Its integral over frequency is assumed to be 128 normalized to unity. For spectral lines, ϕ(x) is in general a Voigt 129

function. The source function S (τ, x) is given by 130

The notation C | /IR stands for the complex plane cut along the full real axis

sion of N(z, ξ) given in Eq. (81). This leads to

Article number, page 11 of 16 A&A proofs: manuscript no. mixture-aa-general

Acknowledgements. The author is very grateful to Dr. M. Hemsch for having introduced her to this radiative transfer problem with partial frequency redistribution and for having provided a copy of his PhD thesis. She is also very thankful for stimulating discussions.

Appendix A.1.

374

Since Ω(z) has no zero, we can divide Eq. ( 47) by Ω(z). Mak-375 ing use of Eq. ( 48), we obtain for N∞ (z) an explicit expression, 376 analytic in C | /IR as it should be, which may be written as 377

1

We can now obtain closed form expressions for N ∞ (z, ξ) and

By construction F(z) is analytic in the complex plane cut along 638 [0, ∞[ and according to Eq. ( 99) its jump along this cut is zero.

639

Hence it is an entire function. It is also easy to see that it goes to 640 zero at infinity. We can thus conclude that F(z) is zero for all z.

641

We thus obtain 642

Applying the Plemelj formulae to all the terms in this equation 643 and using the expression of ρ(ν) given in Eq. ( 92), we obtain for 644 (ν) a Fredholm integral equation, which may be written as

and η(ν) by

An explicit expression of V(ν, ν ′ ) is given in Appendix B and 648 g(ν) is defined in Eq. ( 93). Using the definition of X± (ν) in 649

Eq. ( 94) and the factorization relation in Eq. ( 96), it is possi-650 ble to express all the square brackets in terms of X(-ν) and of 651 the imaginary and real parts of Ω + (ν).

652

Equation ( 102) holds for the Green function, provided the 653 right-hand side is conveniently modified. We now have 654

with

The function G g (z) is a Hilbert transform defined as in Eq. ( 97),

656

the function g(ν) being replaced by

The complete frequency redistribution limits of Eqs. (102)

658

and ( 105) are obtained by setting κ(ν, ν ′ ) to zero and mak-

659

ing appropriate changes to the right-hand sides. In the defin-660 ing equation for η(ν) (see Eq. ( 104)), g(ν) must be replaced by 661 q * (ν)/(ω(1b)). In Eq. ( 106), η g (ν) is reduced to its first term.

662

Indeed, L(z, ξ) = 1 when b = 0, hence its jump is zero. We thus

The Laplace transform of R (ν) yields the resolvent function for

Here we present some exact expressions for an arbitrary primary 723 source Q * (τ) and a ξ dependent branching ratio b.

724

In terms of the source function S (τ, ξ), the emergent intensity 725 is given by 726

We recall that ξ = ϕ(x) and that µ is the cosine of the radiation 727 field direction. The source function is given by Eq. ( 2), its inverse 728 Laplace transform s(ν, ξ) by

and the Hilbert transform of s(ν, ξ) by

Expressing in Eq. ( 115) S (τ, ξ) in terms of its Laplace transform,

731

we can write the emergent intensity as 732

The three functions N(z, ξ), N(z) and N Q (z) are analytic in the 733 complex plane cut along the positive real axis and free of singu-734 larities. The same properties hold true for N S (z, ξ). The emergent 735 intensity may thus be written as

and calculated by a contour integration, with the contour shown

To obtain N S (z, ξ), it suffices to insert into Eq. ( 117) the expres-remarks 793

Although the partial frequency redistribution model considered 794 here is fairly simple, namely a linear combination of monochro-Appendix A: The generalized dispersion function

Ω(z)

The generalized dispersion function Ω(z) is defined in the text by

This definition is used in Section A.1 to analyze the properties of Ω(z). In Section A.2, it used to show that Ω(z) reduces to the complete frequency redistribution dispersion function L R (z) when b = 0. We recall that ξ = ϕ(x), with ϕ(x) the absorption profile and ϕ 0 = ϕ(0).

A.1. Properties of Ω(z)

The properties of Ω(z) are presented here for a constant b. The main ones are unchanged when b is ξ-dependent. The minor changes which are then needed are indicated in Section 5.4.

We first remark that Ω(z) is analytic in the complex plane cut along the real axis, the point zero being excluded. Indeed, it has the same singular line as L(z, ξ), and in the integration over ξ, the lower limit of ξ is zero. For z = 0, we have L(0, ξ) = ǫ b (see Eq. ( 29)). Using L(0, ξ) = 1ω bξT (0, ξ) and the normalization of W(ξ) to unity (see Eq. ( 9)), we readily infer from Eq. (A.1)

that

At infinity, L(z, ξ) → 1 and T (z, ξ) → 0, hence Ω(z) tends to 1.

We now determine for all

of Ω(z), as z tends to the real axis, from above and from below. With the explicit expressions of Ω ± (ν), it will be possible to determine whether or not Ω(z) is free of zeroes. We know that the dispersion function L(ν, ξ) has a continuous spectrum for |ν| > |ξ| and a discrete spectrum for ν ∈ [-ξ, +ξ], with zeroes at ±ν 0 (ξ). The existence of these two different spectra and of these zeroes brings some complexity to the calculation of Ω ± (ν) and the expressions that are obtained depend on the value of ν.

Three different domains shown in Fig. 2 have to be considered. They are referred to as I, for ν larger that ϕ 0 , II, for ϕ 0 u 0 < ν < ϕ 0 , and III, for 0 < ν < ϕ 0 u 0 . Because Ω(z) is an even function of z, we consider only positive values of ν. The straight line ν = ξ separates the continuous from the discrete spectrum. The line ν = u 0 ξ shows the location of the zeroes of L(ν, ξ). We recall that u 0 is the positive zero of L(z) (see Eq. ( 33)) and that u 0 and ϕ 0 are smaller than one.

For all values of ν in this interval, and all values of ξ ∈ [0, ϕ 0 ], ν belongs to C(ξ), that is to the continuous spectrum. The Plemelj formulae can be applied to Eq. (A.1). Using the expressions of L ± (ν) and T ± (ν) given in Eq.( 27) to calculate the jump of T (z, ξ)/L(z, ξ), we obtain

and

We thus obtain

For all ξ < ν, ν belongs to the continuum spectrum and for 1023 all ξ > ν, to the discrete spectrum. In the latter region, L(z, ξ) 1024 being analytic, its jump across the real axis is zero. In domain 1025 II there is no contribution from the zero at ν 0 (ξ) since ν > u 0 ϕ 0 1026 (see Fig. 2). We thus obtain 1027

In the imaginary part, which comes from the continuous 1028 spectrum, the upper limit of the integral is now ν. The situation similar to that of the domain II, but for a given 1033 value of ν = ℜ(z), L(z, ξ) will have a zero when ξ has the value 1034 ξ 0 = ν/u 0 (see Section 3). There is thus in Eq. (A.9) an additional 1035 contribution coming from the integral

The analyticity of T (z, ξ) and L(z, ξ) for ℜ(z) ∈ [-ξ, +ξ], allows 1037 us to deform the integration line in such a way that it will pass 1038 around the pole. The contribution of the pole is found, in the 1039 standard way, by introducing a semi-circle around the pole at 1040 ξ 0 = ν/u 0 , placed below the integration line for Ω + (ν) and above 1041 it for Ω -(ν). Taylor expanding λ(ν, ξ) around ξ 0 , we find that 1042 the contribution of the pole to the integral in Eq. (A.10) may be 1043 written as .11) where ∂ ξ λ(ν, ξ 0 ) is the partial derivative of λ(ν, ξ) with respect 1045 to ξ taken at ξ 0 . This partial derivative has been calculated in 1046 Section 3 (see Eq. ( 35)). We thus find 1047

. (A.12)

A general theory for the calculation of integrals with a pole in 1048 the denominator as in Eq. (A.10) can be found in [START_REF] Gakhov | Boundary Value Problems[END_REF]Gakhov ( , 1049 p. 66-68) and in [START_REF] Hemsch | Radiative transfer with partially coherent scattering[END_REF].

1050

Article number, page 14 of 16

Hélène Frisch: Exact solutions for radiative transfer with partial frequency redistribution

We now turn to the determination of the number of zeroes of Ω(z). A standard method for an analytic function is to examine the variation of its argument along a close contour inside the analyticity region [START_REF] Carrier | Functions of a Complex Variable[END_REF]Case & Zweifel 1967, e.g.,). For Ω(z), which is an even function of z, it suffices to calculate the variation of the argument of Ω + (ν) as ν varies along the positive real axis. A phase diagram of Ω + (ν), show-

A.2. The complete frequency redistribution limit

We verify in this appendix that Ω(z) = L R (z) for b = 0, with L R (z) the dispersion function for complete frequency redistribution. For complete frequency redistribution, the source function depends on τ only and satisfies a convolution-type integral equation, which may be written as

where D is the integration domain. The subscript R stands for complete frequency redistribution. The kernel is given by

It may also be written as

where .17) with ϕ -1 the inverse function of ϕ. The dispersion function is 1084 then given by

The integral in the right-side is the complex Fourier transform of 1086 the kernel.

1087

The frequency dependent dispersion function L(z, ξ), re-1088 duces to L(z, ξ) = 1 when b = 0 (see Eq. ( 22)). Hence Ω(z) 1089 reduces to 1090 we can rewrite Eq. (A.19) as

1095

Interchanging the order of the integrations between ν and x and 1096 using Y(νϕ(x)) = 1, we obtain

1097

The expression of k R (ν) given in Eq. (A.16) immediately shows 1098 that Ω(z) = L R (z).

1099 With a similar calculation, one can verify that the expres-1100 sions given in Section A.1 for the jump Ω + (ν) -Ω -(ν) across the 1101 real axis are identical to the jump of L R (z), namely that we have 1102 +∞[, (A.25) with k R (ν) given by Eq. (A.16). In Domain I y(ν) = 0 and in 1103 domains II and III y(ν) = ϕ -1 (ν), with ϕ -1 the inverse function 1104 of ϕ. Appendix B: The functions M(z, ξ) and V(ν, ν ′ )

1106

The function M(z, ξ) is defined by

where

This double integral can be transformed into the sum of two sim-

and changing the order of integration, we find

A similar equation holds when ν ′ is replaced by a point z in the 1115 complex plane. Summing the two equations, we obtain

We now use the factorization X * (ν, ξ)X * (-ν, ξ) = λ(ν, ξ), which holds for ν ∈ [0, ξ], to transform the second integral in the right-hand side of Eq. (B.7). To calculate M 1 (z, ξ), we use the factorization

In Section 5.3 we show that ρ(ν), the jump of R(z), can be written as

where V(ν, ν ′ ) is the jump across the positive real axis of the function

r(z, ν, ξ) dξ.

(B.10)

The jump of V(z, ν) can be derived from the jump of 1/L(z, ξ), since r(z, ν, ξ) is analytic in the complex plane cut along the negative real axis (see the definition of r(z, ν, ξ) in Eq. ( 84)). To calculate V(ν, ν ′ ) we can thus proceed exactly as for the calculation of the jump of Ω(z) in Appendix A.1.

In domain I (see Fig. 2), corresponding to ν > ξ, the Plemel formulae for L(z, ξ) given in Eq. ( 27) lead to .11) with k(ν, ξ) = 1/(2ν). In domain II, we can use Eq. (B.11) in the continuous part of the spectrum corresponding to ν > ξ but also in the discrete part corresponding to ν < ξ since the jump of L(z, ξ) is zero. In domain III there is a contribution from the zero of L(z, ξ) at ν 0 (ξ) (see Fig. 2), which can be calculated as described in Section A.1.

We can thus write ρ(ν) as in Eq. (B.9) with Here, ξ 0 = ν/u 0 , where u 0 is the positive zero of the function L(z) introduced in Eq. ( 33). In the complete frequency redistribution limit, V(ν, ν ′ ) and ρ(ν) are zero.

Article number, page 16 of 16