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# Relations between solid harmonics Rn,m and Sn,m

Noting the following formulae (See Hobson[43] or Wallace[80]):

 (B.1)

 (B.2)

 (B.3)

 (B.4)

one can transform as follows:

where we have used (A.5) and the chain rule:

The underlined part (a) can be rewritten as follows:
In the case where

In the case where m' < m

As a result, the underlined part (a) can be rewritten as

Hence one obtains

 = = = (B.5)

Now take two points O and O' where two inequalities and are valid and consider the following identity:

 (B.6)

From (B.5) we obtain the following formula:

 (B.7)

Substituting (B.7) into the left-hand side of (B.6), one obtains

 (B.8)

Setting n+n'=n'',m+m'=m'' and deleting n in (B.8), one has

 = = (B.9)

Exchanging the order of the summation in (B.9) ( , See Fig.B), one obtains

 = = = (B.10)

Finally, comparing the right-hand side of (B.6) with that of (B.10) one obtains the following formula:

 (B.11)

Next: Recurrence formulae for Rn,m Sn,m Up: Applications of Fast Multipole Previous: Series expansion of
Ken-ichi Yoshida
2001-07-28