where is the multipole moment centred at

In (3.30) we have assumed that the inequality ( ) holds. The multipole moment is transformed when the centre of the multipole moment is translated from

where we have used (3.21) and (3.31). has the same property as that for

The integral in (3.30) can be evaluated with the coefficient of the local expansion as follows:

where is expressed in terms of by

In the derivation of this formula we have used (3.20) and have assumed that the inequality holds. Also, has the same property as that for given in (3.28):

The coefficient of the local expansion is translated according to the
following formula when the centre of the local expansion is shifted
from
to

where we have used (3.21) and (3.34).

Notice that in FM-BIEM with the regularised integral equation the
multipole moments and the coefficients of the local expansion have three
components for a given pair of *n* and *m*. This means that this
formulation requires three times as much computational cost for M2M, M2L
and L2L translation as FM-BIEM with the hypersingular integral equation
obtained in the previous section and moreover the memory requirement of
the regularised formulation is also triple that of the hypersingular
one. However, because in the regularised formulation we can compute
the multipole moments analytically (See the following section), FM-BIEM
with regularisation can be expected to yield more accurate results than
that without regularisation. Hence, we investigate the difference
between numerical results obtained with two formulation through
numerical experiments.

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.