where , denotes the complex conjugate and the functions and are defined as

In these formulae and are the spherical Bessel function and the spherical Hankel function of the first kind, respectively, indicates the unit vector given by and the spherical harmonics is defined as

The functions and have the following properties (See Epton and Dembart[17]):

where

In (3.125) stands for the Wigner-3j symbol (See Messiah[59]). The explicit form of the Wigner-3j symbol is expressed as

The summation is over such

In this section we use the hypersingular integral equation
(3.118) for the formulation. We now compute the integral on the
right-hand side of (3.118) over a subset of *S* denoted by
*S*_{y} for
which is away from *S*_{y}. Using
(3.120), we obtain

where and

In (3.127) we have assumed that the inequality ( ) holds.

The multipole moments are translated according to the following
formulae as the centre of the multipole expansion is shifted from *O* to
*O*':

where we have used (3.123) (See Appendix I.1). In the evaluation of the integral on the right-hand side of (3.118) one can use not only the multipole moments but also the coefficients of local expansion in the following manner:

where , is expressed with

and is defined as

In the derivation of (3.131) we have used (3.124) and have assumed that the inequality holds.

The coefficients of the local expansion are translated according to the
following formulae as the centre of the local expansion is shifted from
to
:

where we have used (3.123) (See Appendix I.3).

Notice that in this formulation the multipole moment and the coefficient do not have the properties like those of (3.23) and (3.27) in Laplace's equation because is complex-valued. Therefore, we can not save the memory with the use of symmetries.

We have thus prepared all the formulae needed for FM-BIEM. Using formulae presented above and the algorithm described in chapter 2, one can implement FM-BIEM.