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M2M translation

Suppose that one can expand the function $\psi_i$ as follows:
$\displaystyle \psi_j(\overrightarrow{O'\mbox{\boldmath$ x $ }})=\sum_{i}\psi_i(\overrightarrow{O\mbox{\boldmath$ x $ }}) \alpha_i^j(\overrightarrow{OO'}),$     (2.5)

where $\alpha_i^j$ is the coefficient of the expansion.

The multipole moment centred at O' is given by

$\displaystyle M_j(O') = \sum_{L=1}^{N} \int_{S_L} \psi_j(\overrightarrow{O'\mbox{\boldmath$ y $ }}) \varphi(\mbox{\boldmath$ y $ }) dS.$     (2.6)

Substituting (2.5) into (2.6) one obtains the following formula:
$\displaystyle M_j(O') = \sum_{i} M_i(O) \alpha_i^j(\overrightarrow{OO'}).$     (2.7)

This formula (2.7) is used to translate the multipole moment as the centre of the multipole expansion is shifted from O to O' (See Fig.2.5) and this translation is called ``M2M (Multipole moment to(2) Multipole moment) translation''. Notice that one can evaluate the integral in (2.1) with the translated multipole moments as follows:

\begin{eqnarray*}\int_{S_y} K(\mbox{\boldmath$ x $ },\mbox{\boldmath$ y $ }) \va...
..._{i} \phi_i(\overrightarrow{O'\mbox{\boldmath$ x $ }}) M_i(O').

Ken-ichi Yoshida