Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
and then from Eq. (7.96) we obtain
Z 1
0
xJ n …x†J n …x†dx ˆ 0:
…7:98†
p
p
Thus, the two functions x Jn …x† and
x Jn …x† are orthogonal in (0, 1). We can
also say that the two functions J n …x† and J n …x† are orthogonal with respect to
the weighted function x.
Eq. (7.98) is also easily proved if R ˆ 0 and S 6ˆ 0, or R 6ˆ 0 but S ˆ 0. In this
case, and can be any two di€erent roots of J n …x† ˆ0orJn…x† 0 ˆ0.
Spherical Bessel functions
In physics we often meet the following equation
d
r 2 dR
‡‰k 2 r 2 l…l ‡ 1†ŠR ˆ 0;
dr dr
…l ˆ 0; 1; 2; ...†: …7:99†
In fact, this is the radial equation of the wave and the Helmholtz partial di€erential
equation in the spherical coordinate system (see Problem 7.22). If we let
x ˆ kr and y…x† ˆR…r†, then Eq. (7.99) becomes
x 2 y 00 ‡ 2xy 0 ‡‰x 2 l…l ‡ 1†Šy ˆ 0 …l ˆ 0; 1; 2; ...†; …7:100†
where y 0 ˆ dy=dx. This equation almost matches Bessel's equation (7.71). Let us
make the further substitution
p
y…x† ˆw…x†=
x ;
then we obtain
x 2 w 00 ‡ xw 0 ‡‰x 2 …l ‡ 1 2
†Šw ˆ 0 …l ˆ 0; 1; 2; ...†: …7:101†
The reader should recognize this equation as Bessel's equation of order l ‡ 1 2 .It
follows that the solutions of Eq. (7.100) can be written in the form
y…x† ˆA J l‡1=2…x†
p ‡ B J l1=2…x†
p :
x
x
p
This leads us to de®ne spherical Bessel functions j l …x† ˆCJ l‡E …x†=
p
x . The factor
C is usually chosen to be =2 for a reason to be explained later:
p
j l …x† ˆ
=2x J l ‡ E…x†:
…7:102†
Similarly, we can de®ne
p
n l …x† ˆ
=2x N l‡E …x†:
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