Mathematical Methods for Physicists: A concise introduction - Site Map

BESSEL'S EQUATION
relation (7.77) breaks down when 2 ˆ k and a second solution has to be found
by other methods. There is a diculty also when ˆ 0, in which case the two
roots of the indicial equation are equal; the second solution must also found by
other methods. These will be discussed in next section.
The results of Problem 7.16 are a special case of an important general theorem
which states that J …x† is expressible in ®nite terms by means of algebraic and
trigonometrical functions of x whenever is half of an odd integer. Further
examples are
J 3=2 …x† ˆ 2 1=2
sin x
cos x ;
J 5=2 …x† ˆ 2
x
x x
1=2
3 sin x
x
‡ 3
x 2 1 cos x
The functions J …n‡1=2† …x† and J …n‡1=2† …x†, where n is a positive integer or zero, are
called spherical Bessel functions; they have important applications in problems of
wave motion in which spherical polar coordinates are appropriate.
:
Bessel functions of the second kind Y n …x†
For integer ˆ n; J n …x† and J n …x† are linearly dependent and do not form a
fundamental system. We shall now obtain a second independent solution, starting
with the case n ˆ 0. In this case Bessel's equation may be written
xy 00 ‡ y 0 ‡ xy ˆ 0;
…7:80†
the indicial equation (7.73a) now, with ˆ 0, has the double root ˆ 0. Then we
see from Eq. (7.33) that the desired solution must be of the form
y 2 …x† ˆJ 0 …x† ln x ‡ X1
A m x m :
Next we substitute y 2 and its derivatives
mˆ1
…7:81†
y 0 2 ˆ J 0
0 ln x ‡ J 0
x ‡ X1
mˆ1
mA m x m1 ;
y2 00 ˆ J0 00 ln x ‡ 2J 0
0
x J 0
x 2 ‡ X1
mˆ1
m…m 1†A m x m2
into Eq. (7.80). Then the logarithmic terms disappear because J 0 is a solution of
Eq. (7.80), the other two terms containing J 0 cancel, and we ®nd
2J 0
0 ‡ X1
mˆ1
m…m 1†A m x m1 ‡ X1
mˆ1
mA m x m1 ‡ X1
mˆ1
A m x m‡1 ˆ 0:
325