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Derivatives of
R_{n,m}
and S_{n,m}
We introduce a vector
which has components
(x_{1},x_{2},x_{3}) in
the Cartesian coordinates and
in the polar
coordinates. Relations between
(x_{1},x_{2},x_{3}) and
are expressed as
According to the chain rule, we have the following relations:
the coefficients of which are given as
Now we compute
for example.
where we have used standard formulae given by
In the same way we obtain formulae given by

= 
R_{n1,m1}, 
(D.2) 

= 
R_{n1,m}. 
(D.3) 
We can rewrite (D.1), (D.2), (D.3) as

= 

(D.4) 

= 

(D.5) 

= 
R_{n1,m}. 
(D.6) 
Derivatives of S_{n,m} can be obtained in the similar manner as
follows:

= 
 S_{n+1,m+1}, 
(D.7) 

= 
S_{n+1,m1}, 
(D.8) 

= 

(D.9) 

= 

(D.10) 

= 
 S_{n+1,m}. 
(D.11) 
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Up: Applications of Fast Multipole
Previous: Recurrence formulae for R_{n,m} S_{n,m}
Kenichi Yoshida
20010728