Spherical Bessel and Hankel functions

The problem is not simplified by Eq. (15), since there exists a closed-form expression for the multi-scattering matrix for n spheres in terms of spherical Bessel and Hankel functions, spherical harmonics and 3j-symbols, where l, l and to, m are total angular momentum and z-projection quantum numbers, respectively (Henseler, Wirzba and Guhr, 1997) ... 

Taking into account the properties of spherical harmonics [70], Clebsch-Gordon coefficients [71], and spherical Bessel and Hankel functions [70], it is possible to show that the mode functions in (18) obey the following condition of symmetry ... 

In this appendix we recall the basic properties of the solutions to the Bessel and Legendre differential equations and discuss some computational aspects. Properties of spherical Bessel and Hankel functions and (associated) Legendre functions can be found in [1,40,215,238]. 

The solution of these equations is represented by certain combinations of spherical Bessel or Hankel functions and spherical harmonics [24—26]. 

The functions PJT(cos 9) are associated Legendre functions of the first kind of degree n and order m, and z (kr) denotes any of four spherical Bessel functions. The choice of the spherical Bessel function depends on the domain of interest, that is, on whether we are looking for the solution inside the sphere (r < a) or outside the sphere (r > a). For the internal field we choose z (kr) = j (kr), where j (kr) is the spherical Bessel function of the first kind of order n. The solution for the external field can be written in terms of spherical Bessel functions j kr) and y kr), where the latter is the spherical Bessel function of the second kind, but it is more convenient to introduce the spherical Hankel function /i / (kr) to determine tj/ for the outer field. 

These results can be put in a more useful and simpler form if kr is sufficiently large to permit asymptotic forms of the spherical Bessel functions and spherical Hankel functions to be applied. In this case the transverse components of the scattered electric vector are... 

Any linear combination of jn and yn is also a solution to (4.5). If the mood were to strike us, therefore, we could just as well take as fundamental solutions to (4.5) any two linearly independent combinations. Two such combinations deserve special attention, the spherical Bessel functions of the third kind (sometimes called spherical Hankel functions) ... 

They are complemented by the Hankel functions of first and second kind (spherical Bessel function of third and fourth kind)... 

For the general description of wave fields it is important to note that the spherical Bessel functions of first kind j diverges at small arguments and vanishes at infinity, while the opposite applies to the spherical Hankel functions of first kind... 

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