Geoscience Reference
In-Depth Information
1.0
Q
n
(t
Q
0
Q
2
0.5
0
-1.0
-0.5
0.5
1.0
-0.5
Q
4
-1.0
t=cos
#
1.0
Q
n
(t
Q
3
0.5
Q
5
0
-1.0
-0.5
0.5
1.0
-0.5
Q
1
-1.0
t=cos
#
Fig. 1.6. Legendre's functions of the second kind:
n
even (top) and
n
odd (bottom)
kind as functions of a complex argument. If the argument
z
is complex, we
must replace the definition (1-74) by
n
2
P
n
(
z
)ln
z
+1
Q
n
(
z
)=
1
1
k
P
k−
1
(
z
)
P
n−k
(
z
)
,
1
−
(1-77)
z
−
k
=1
where Legendre's polynomials
P
n
(
z
) are defined by the same formulas as in
the case of a real argument
t
. Therefore, the only change as compared to
(1-74) is the replacement of
1
2
ln
1+
t
1
t
=tanh
−
1
t
(1-78)
−
by
1
2
ln
z
+1
z
1
=coth
−
1
z.
(1-79)
−
