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M2L translation formula

Substituting (3.124) into (3.127), we obtain

\begin{eqnarray*}\lefteqn{\sum_{n=0}^{\infty}\sum_{m=-n}^{n} (2n+1)
\frac{\parti...
..._0\mbox{\boldmath$ x $ }})L_{n}^{m}(k,\mbox{\boldmath$ x $ }_0),
\end{eqnarray*}


where $L_n^m(k,\mbox{\boldmath$\space x $ }_0)$ is the coefficient of the local expansion at $\mbox{\boldmath$\space x $ }_0$ given by
 
$\displaystyle L_n^m(k,\mbox{\boldmath$ x $ }_0)=\sum_{n'=0}^{\infty}\sum_{m'=-n...
...',m,l}
\widetilde{{\cal O}}_l^{-m-m'}(k,\overrightarrow{Ox_0})M_{n'}^{m'}(k,O).$     (I.2)

In the derivation of (I.2) we have used the following identities:

\begin{eqnarray*}Y_n^m(\widehat{O\mbox{\boldmath$ x $ }}) = (-1)^n Y_n^m(\wideha...
...}) = (-1)^m \overline{Y}_n^m(\widehat{O\mbox{\boldmath$ x $ }}).
\end{eqnarray*}




Ken-ichi Yoshida
2001-07-28