Geoscience Reference
In-Depth Information
Appendix A: Spherical Bessel Functions
The spherical Bessel functions are the solutions of the second order differential
equation (Abramowitz and Stegun
1964
)
z
2
w
0
n
C
2
zw
0
n
C
z
2
n.n
C
1/
w
n
D
0;
(4.53)
where
z
and
w
.
z
/ are generally complex and the prime denotes the derivative with
respect to
z
. The spherical Bessel functions of the third kind are the partial and
linearly independent solutions of Eq. (
4.53
). They can be expressed as follows:
h
.
1
n
.
z
/
D
i
z
n
n
exp .i
z
/
z
;
d
z
d
z
h
.2
n
.
z
/
D
i
z
n
n
exp .
i
z
/
z
;
d
z
d
z
(4.54)
where n
D
0;1;2;:::
Legendre Polynomials
The Legendre polynomials, P
n
.cos /, are the solutions of the differential equation
d
.
sin d
P
n
/
sin
C
n.n
C
1/P
n
D
0;
(4.55)
where d
D
d=d denotes the derivative with respect to . These polynomials are
defined as follows:
.d cos /
n
cos
2
1
n
:
d
n
1
2
n
nŠ
P
n
.cos /
D
(4.56)
The Legendre polynomials are mutually orthogonal functions on the segment
Œ
;, that means that
Z
P
n
.cos /P
m
.cos / sin d
D
0; if n
¤
m;
(4.57)
0
and
Z
2
2n
C
1
:
ŒP
n
.cos /
2
sin d
D
(4.58)
0
Search WWH ::
Custom Search