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Expansion of a kernel function

In FM-BIEM the first step is to expand the kernel function into a series of products of functions of $\mbox{\boldmath$\space x $ }$ and those of $\mbox{\boldmath$\space y $ }$ as follows:
$\displaystyle K(\mbox{\boldmath$ x $ },\mbox{\boldmath$ y $ })=\sum_{i}\phi_i(\...
...row{O\mbox{\boldmath$ x $ }}) \psi_i(\overrightarrow{O\mbox{\boldmath$ y $ }}),$     (2.2)

where $\psi_i$ is regular near the origin O, $\phi_i$ is regular at infinity and O is a certain point such that the inequality $\vert\overrightarrow{O\mbox{\boldmath$\space y $ }}\vert
< \vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert$ is valid (See Fig.2.5).
Figure 2.5: Configuration of points and regions
\epsfile{file=FIG/fmbiem2.eps,scale=0.9} \end{center} \end{figure}

Ken-ichi Yoshida