Next: Series expansion of the
Up: Applications of Fast Multipole
Previous: L2L translation formula
Series expansion of xy
We introduce two vectors
and
which satisfy the
inequality:
.
An expansion for the fundamental solution of threedimensional
Helmholtz's Equation[17] is given by



(F.1) 
where
and
are unit vectors defined as
j_{n} is the spherical Bessel function, h_{n}^{(1)} is defined with the spherical Bessel
functions j_{n} and y_{n} as
and j_{n} and y_{n} are expressed in Taylor (Laurent) series as
j_{n}(kz) 
= 

(F.2) 
y_{n}(kz) 
= 

(F.3) 
We next compare the coefficient of k in the lefthand side of
(F.1) with that in the righthand side of (F.1) to
obtain a series expansion of
.
 the coefficient of k in the lefthand side of (F.1)
The lefthand side of (F.1) is expanded into a power series of
wave number as



(F.4) 
and therefore the coefficient of k in the lefthand side of
(F.1) is



(F.5) 
 the coefficient of k in the righthand side of (F.1)
To begin with, we note that the coefficient of kin
is zero. The coefficient of k in
is
Hence the coefficient of k in the righthand side of (F.1) is







(F.6) 
Comparing (F.5) with (F.6), we obtain a series expansion of
given as
where
and
are functions defined
as



(F.9) 
Hayami and Sauter [41] obtained the same series expansion of
in another way.
Next: Series expansion of the
Up: Applications of Fast Multipole
Previous: L2L translation formula
Kenichi Yoshida
20010728