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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
mλ
and sin
mλ
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