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X2X translation formula

Shifting the centre of the exponential expansion from O to $\mbox{\boldmath$\space x $ }_0$we have the following identity
$\displaystyle {\sum_{k=1}^{s}\sum_{j=1}^{M(k)}
e^{-(\lambda_k/d) ((\ov...
...lpha_j(k) - i (\overrightarrow{O\mbox{\boldmath$ x $ }})_2 \sin \alpha_j(k))} }$
    $\displaystyle =\sum_{k=1}^{s}\sum_{j=1}^{M(k)}
X(k,j;\mbox{\boldmath$ x $ }_0)
...ghtarrow{\mbox{\boldmath$ x $ }_0\mbox{\boldmath$ x $ }})_2 \sin \alpha_j(k))},$ (J.3)

where $X(k,j;\mbox{\boldmath$\space x $ }_0)$ is the exponential expansion centred at $\mbox{\boldmath$\space x $ }_0$defined as

\begin{eqnarray*}X(k,j;\mbox{\boldmath$ x $ }_0) = X(k,j;O)
e^{-(\lambda_k/d) (...
...verrightarrow{O\mbox{\boldmath$ x $ }_0})_2 \sin \alpha_j(k))}.

The computational cost for X2X translation is obviously $O(S\mbox{exp})
\approx O(p^2)$.

Ken-ichi Yoshida