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We can consider Laplace's equation to be the special case Helmholtz's
equation as k tends to 0. Therefore one expects that the FMM
formulation for Laplace's equation is obtained as the limit of
in the FMM formulation for Helmholtz's equation.
However, the formulation obtained above does not allow such operation in
the form as it is. It is because the behaviours of
and
near k=0 are
O(k^{n1}) and
O(k^{n}) and, hence,
In order to obtain a formulation allowing an obvious transition
between these governing equations we need some modifications to cancel
these behaviours.
Noting the power series of
and
,
we obtain
the following formulation



(3.134) 
where
and
are
defined as
Since
the expression in (3.135) is seen to approach (3.120)
as
.
Next: Threedimensional scattering of elastic
Up: Threedimensional scattering of scalar
Previous: Many pennyshaped cracks
Kenichi Yoshida
20010728