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Truncation of the infinite series

In FM-BIEM, we truncate the infinite series in (3.146), (3.151), etc. via (3.133). For example we may truncate (3.146) as follows:
 
$\displaystyle {\int_{S_y}
C_{jktp} \frac{\partial^2}{\partial x_{l}\partial
y_{...
... y $ }) n_{k}(\mbox{\boldmath$ y $ })
\phi_{j}(\mbox{\boldmath$ y $ }) dS_{y} }$
  = $\displaystyle \sum_{n=0}^{P_T} \sum_{m=-n}^{n}
\frac{\partial}{\partialx_l} D^{...
... D^{L,{\cal O}}_{i,n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})
M^L_{n,m}(O),$ (3.154)

where PT and PL are given by (See (3.133))

\begin{eqnarray*}P_T = P(k_T R),\quad P_L = P(k_L R),
\end{eqnarray*}


and the inequality $P_T > P_L (\raisebox{1ex}{.}.\raisebox{1ex}{.} {k_T > k_L})$ holds. However, if one uses (3.156) in this manner the implementation becomes somewhat complicated. Therefore, in the present implementation we truncate the right-hand side of (3.146) taking PT terms for simplicity as follows:

\begin{eqnarray*}\lefteqn{\int_{S_y}
C_{jktp} \frac{\partial^2}{\partial x_{l}\...
...(\overrightarrow{O\mbox{\boldmath$ x $ }})
M^L_{n,m}(O)\right).
\end{eqnarray*}


In the same way we treat other infinite series in (3.151), etc.

Ken-ichi Yoshida
2001-07-28