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and so on.
The geometrical representation of these spherical harmonics is useful.
The harmonics with m = 0 - that is, Legendre's polynomials - are polyno-
mials of degree n in t , so that they have n zeros. These n zeros are all real
and situated in the interval
π (Fig. 1.4).
Therefore, the harmonics with m = 0 change their sign n times in this inter-
val; furthermore, they do not depend on λ . Their geometrical representation
is therefore similar to Fig. 1.5 a. Since they divide the sphere into zones, they
are also called zonal harmonics .
The associated Legendre functions change their sign n
1
t
+1, that is, 0
ϑ
m times in the
interval 0
π . The functions cos and sin have 2 m zeros in the
interval 0 ≤ λ< 2 π , so that the geometrical representation of the harmonics
for m
ϑ
= 0 is similar to that of Fig. 1.5 b. They divide the sphere into com-
partments in which they are alternately positive and negative, somewhat like
a chess board, and are called tesseral harmonics . “Tessera” means a square
or rectangle, or also a tile. In particular, for n = m , they degenerate into
functions that divide the sphere into positive and negative sectors, in which
case they are called sectorial harmonics , see Fig. 1.5 c.
P 6 (cos
#
P 12,6 (cos
#
)cos 6
¸
P 6,6 (cos
#
)cos 6
¸
(a)
(b)
(c)
Fig. 1.5. The kinds of spherical harmonics: (a) zonal, (b) tesseral, (c) sectorial
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