Geoscience Reference
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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
P n .cos / D
(4.56)
The Legendre polynomials are mutually orthogonal functions on the segment
Π;, 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
 
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