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Series expansion of
Now we take two points
and
whose spherical
coordinates are
and
.
The inverse of the distance between
and
can be expanded into
the following series[61]

= 

(A.1) 
where
is the Neumann factor;
and P_{n}^{m}(x) is the associated Legendre
function. In this thesis the associated Legendre function is
defined by
where P_{n}(x) is the Legendre function and the associated Legendre
function has the following properties:
The righthand side in (A.1) can be rewritten as
where we have used (A.3) and functions
and
are defined by
The solid harmonics R_{n,m} and S_{n,m} have the following properties.
where we have used (A.3), and
where we have used (A.4).
Next: Relations between solid harmonics R_{n,m} S_{n,m}
Up: Applications of Fast Multipole
Previous: Final remarks
Kenichi Yoshida
20010728