1. Introduction

In our first accompanying paper we have derived Case’s singular eigenfunctions (CSEs) solution of the light transport equation in the infinite homogeneous tissue with m-dependent anisotropic scattering kernel and benchmarked four computational methods for the m-dependent Chandrasekhar orthogonal polynomials. In this paper we presented the corresponding Fourier transform solution and then showed the consistence of the solutions obtained from the two approaches.

The CSEs solutions are always seen as the exact solutions of integro-differential transport equations and naturally suitable for the light transport in turbid tissue [1–9]. Fourier transform method is a useful method and applied successfully to the transport equations in the regular space [10–17]. Ganapol [12–16] derived the Fourier transform solution for the infinite medium with azimuthal symmetric scattering kernel and showed the consistent theory in a more mathematically unified manner. Machida [17] used the Fourier transform and rotated reference frames technique to compute three-dimension Green’s function for three-dimension linear Boltzmann equation. Their outstanding work opened up a new scope to solve the exact radiance distributions that we are seeking for the light transport in biological tissue.

Firstly, by introducing the associated Chandrasekhar polynomials of the second kind, the general analytical solutions for three-term recurrence in Fourier k-domain are extended to the azimuth-dependent or m-dependent case explicitly as the base of the Fourier transform solutions. Secondly, in m-dependent case, taking the associated Chandrasekhar polynomials of the first/second kind as the Legendre moments, the generalized singular eigenfunctions of the first/second kind (GSE1, 2) are defined, in the limit version and in the truncated version, in terms of the associated Legendre polynomials of the first kind with the same degrees and orders. And also, the implicit corresponding relations between GSEs and CSEs are pointed out and proven from the perspective of both expression forms and intrinsic behaviors. Depending on the first Legendre moment, the general analytical solution can be expressed completely and used to Fourier inversion. Lastly, in the Fourier inversion, the contributions from the poles and the branch cut are combined to the final Fourier transform solution. The poles in complex plane correspond to the discrete values in Case’s method and the branch cut corresponds to the continuous spectrum, and then the final solution confirms perfectly to the CSEs solution.

During the derivation, many identities are different from those in the case $m=0$, such as the Christoffel-Darboux (C-D) identities or Liouville-Ostrogradski (L-O) formulas [14–19], ${\Lambda }_L^m$ etc. The derivations of these identities are lengthy but very helpful to future study.

2. M-dependent polynomials and C-D formulas

In our following derivations, many m-dependent polynomials or functions and corresponding C-D formulas will be used frequently. However, there are some aspects different from that of the case$m=0$. In this section, we give brief reviews of these polynomials and formulas.

In the real domain [20–22], $P_l\left(\mu \right)\left(-1\le \mu \le 1\right)$ is the Legendre polynomial of the first kind of degree$l$ . $P_l^m\left(\mu \right)\left(-1\le \mu \le 1,l\ge m\ge 0\right)$ is the associated Legendre polynomial of the first kind of order $m$ and degree $l$, and the Hobson’s definition is

In the complex plane, $P_l\left(z\right)$ is a Legendre orthogonal polynomial [20] and $P_l^m\left(z\right)$is defined as [20–22]

_{$Q_l\left(z\right)$} is the Legendre function of the second kind and defined in the complex plane excluding a cut line $-\infty$ to 1. However, in our research we usually use the following definition

The Chandrasekhar polynomials $g_l^m\left(z\right)$ and ${\rho }_l^m\left(z\right)$are defined in our first accompanying paper and can be extended into the complex plane. They satisfy the three-term recurrence relations

The Christoffel-Darboux (C-D) identities or Liouville-Ostrogradski (L-O) formulas about above polynomials are ordinary and not difficult to derive, however, in our case$m>0$, there are still some points that deserve careful consideration. For example, the definitions and three-term recurrence of $Q_l^m\left(z\right)$ should be noticed during the derivation. We have already derived these formulas, and listed here for future use. In the following formulas, $L$is the truncated position of the phase function, that is to say when$l>L$,$f_l=0$. $N$ is an integer, and$N>L$.$\varpi ={\mu }_s/{\mu }_t$ is the single particle albedo, where ${\mu }_s$is the scattering coefficient, ${\mu }_a$ is the absorption coefficient, and ${\mu }_t={\mu }_a+{\mu }_s$ is the total attenuation coefficient.

3. Standard Fourier transform and matrix inversion

Let’s begin with the Eq. (4) in our first accompanying paper:

Then, we introduce the Fourier transform and inversion

Expanding the ${\tilde{\psi }}^m\left(k,\mu ;{\mu }_0\right)$ by the associated Legendre polynomials of the first kind

In the derivation from Eq. (27) to Eq. (29), $k$ is constrained in the complex plane excluding cut lines $\left[i,\infty \right)$ and $\left[-i,-\infty \right)$ on the imaginary axis, and correspondingly $z$ is defined in the complex plane excluding a cut line −1 to + 1 on the real axis.

Due to ${\beta }_l^m=0$ for$l>L$, when$m\le j\le L$, Eq. (28) can be rewritten as

Matrix inversion can be directly used to get the moment vector of ${\tilde{\psi }}_{L,l}^m\left(k,\mu ;{\mu }_0\right)$

Then, from Eq. (25), Fourier transform inversion can be applied to Eq. (36) to get the final conventional solution, and we don’t deduce further.

4. Fourier domain solution and GSEs

4.1 Solution of three-term recurrence

From Eq. (4), we can also get the three-term recurrence

We assume that the first Legendre moment of${\tilde{\psi }}^m\left(k;{\mu }_0\right)$,${\tilde{\psi }}_m^m\left(k;{\mu }_0\right)$, is known, then we let $l=m$, $a^m\left(z;{\mu }_0\right)={\tilde{\psi }}_m^m\left(k;{\mu }_0\right)/g_m^m\left(z\right)$, and $b^m\left(z;{\mu }_0\right)=0$in Eq. (38) and get

Substituting Eq. (39) into Eq. (37), to keep balance between the equations, the following relations must be satisfied

Equation (40) is a three-term recurrence, too. Considering that Eq. (40) is a homogeneous equation, the particular solution is zero, and ${\eta }_{l,j}^m\left(z\right)$ can be expressed by general solution form

The coefficients $c_j^m\left(z\right),d_j^m\left(z\right)$ are determined by substituting Eq. (43) into Eq. (41) and Eq. (42)

From Eq. (21), ${\eta }_{l,j}^m\left(z\right)$ now is

Thus, the analytical solution Eq. (37) can be expressed as

_{${\chi }_l^m\left(z,\mu \right)$} is the auxiliary function and${\chi }_m^m\left(z,\mu \right)=0$. From Eq. (48) and Eq. (49), we can see that if the first Legendre moment is known, the other Legendre moments will be easily got. The first Legendre moment plays an important part during the derivation.

4.2 The first Legendre moment

The first Legendre moment is so important that Ganapol had provided several methods of solving it [12–16]. When$m=0$, those methods are all valid. However, when$m>0$, it is impossible to use the isotropic point source to find the first Legendre moment. So, we substitute the Eq. (48) into Eq. (30), let$j=m$, and use

In our first accompanying paper, we defined a holomorphic function in the complex plane excluding a cut line −1 to 1 on the real axis

Substituting$P_m^m\left(z\right)=\left(2m-1\right)!!{\left(z^2-1\right)}^{m/2}$ and $g_m^m\left(z\right)=\left(2m-1\right)!!$ into Eq. (52), it is obvious that Eq. (52) and Eq. (53) are identical.

4.3 The generalized singular eigenfunctions of the first kind

Now, we can begin to build the generalized singular eigenfunctions of the first kind (GSE1) which are extended from the Ganapol’s definition for the case $m=0$ in the complex z-plane [15, 16].

Let’s define GSE1, called the limit version, as the expansion series of the combination of the associated Chandrasekhar polynomials of the first kind and the associated Legendre polynomials of first kind

Substituting Eq. (13) into Eq. (56), we get

Because$l>L$,$f_l=0$, ${\phi }_N^m\left(z,\mu \right)$will remain unchanged for any$N>L$,

From Eq. (56), the analyticity of ${\varphi }_L^m\left(z,\mu ;N\right)$ depends on that of$g_l^m\left(z\right)$. When$l\le L$, as discussed in our first accompanying paper, it is certain that$g_l^m\left(z\right)$ is an analytic function because it can be expressed with polynomials explicitly or a tri-diagonal determinant which can be factorized with the eigenvalues. When$l>L$, from Eq. (9), the three-term recurrence for $g_l^m\left(z\right)$ will be the homogeneous form

Considering the $g_l^m\left(z\right)$ degenerates into associated Legendre polynomials when$l>L$, we can use the form of the general analytical solution of three-term recurrence again

According to the continuity of $g_l^m\left(z\right)$ at the truncated position, we have

Using Eq. (22), the coefficients can be solved

So when$l>L$,

By letting $l=L$or $l=L+1$in Eq. (65), the correctness of the expression of $g_l^m\left(z\right)$ is apparent. However, due to the singularity of$Q_l^m\left(z\right)$, the analyticity of $g_l^m\left(z\right)$ needs to be proven. When$m=0$, Ganapol [15, 16] proved it by the relationship between $Q_l\left(z\right)$ and $P_l\left(z\right)$. In fact, from the definition of $Q_l^m\left(z\right)$ in Eq. (6), we write down the part of Eq. (65)

Because $P_l\left(z\right)$ is a polynomial in $z$ of degree$l$, and $P_l^m\left(\mu \right)$ is a polynomial in $\mu$ of degree$l$, it can predicate that the denominator will vanish and the expression is analytic. So we can conclude that $g_l^m\left(z\right)$ is an analytic polynomials in the complex z-plane and then ${\psi }_L^m\left(z\right)$ is analytic, too. So, ${\varphi }_L^m\left(z,\mu ;N\right)$ is actually analytic function in the complex z-plane. But from Eq. (63), ${\Lambda }_L^m\left(z\right)$ is only analytic excluding a cut line −1 to 1 on the real axis.

Substituting Eq. (64) in Eq. (56), we get the truncated version of GSE1

From Eq. (54) and Eq. (59), ${\phi }_L^m\left(z,\mu \right)$ is identical to$g^m\left(z,\mu \right)$ in Case’s method. Thus, the truncated version of GSE1,${\varphi }_L^m\left(z,\mu ;N\right)$, is expressed by all known functions and similar to the eigenfunctions in Case’s method.

Now we can inspect the relations between the CSEs and GSEs. Let’s check the discrete eigenfunctions first. Because the discrete eigenvalues in Case’s method are real and their absolute values are greater than 1, here we let $z$ is real number, i.e.$v$, and$\left|v\right|>1$. Though $\delta \left(v-\mu \right)$ will vanish for discrete eigenvalues, expanding the Dirac delta function in terms of the associated Legendre polynomials of the first kind, and we get

From Eq. (19), when$z=v$we get

So

It should be noticed that Eq. (70) ~Eq. (72) are only valid in real domain. In complex domain, Ganapol suggested [16] that the Dirac delta function should be viewed as a placeholder and the limit came at the end when $z$ was taken to the cut line to become real.

Expanding $\frac{1}{2}\frac{{\left(1-{\mu }^2\right)}^{m/2}{\left(v^2-1\right)}^{-m/2}}{v-\mu }$ in terms of the associated Legendre polynomials of the first kind, we get