13.7 Bessel functions 387
Both the expansions and the graphs show that the Bessel functions have properties
similar to the trigonometric functions. However, the values of xfor whichJ
n(x) 1 = 10 ,
the zeros of the functions, are not equally spaced so that the functions do not have a
fixed wavelength. Some of the zeros are (to 3 decimal places)
J
0(x) 1 = 1 0forx 1 = 1 2.405, 5.520, 8.654, 11.792, 14.931, = (13.53)
J
1(x) 1 = 1 0forx 1 = 1 0, 3.832, 7.016, 10.173, 13.324, =
In addition, the amplitude of the waves decreases as xincreases. The behaviour for
large values of xis that of a damped sine function,
(13.54)
For positive values of nthe particular solution (13.49) isJ
−n(x)and it can be shown
that this is related toJ
n(x)byJ
−n(x) 1 = 1 (−1)
nJ
n(x). The power-series method therefore
gives only one independent solution for integral values of n. A second solution can be
found; this is the Bessel function of the second kind, Y
n(x), of less importance for
physical applications.
0 Exercise 25
Bessel functions J
l+ 122(x)of half-integer order
Bessel functions of half-integral order can be expressed in terms of elementary
functions. The functions forn 1 = 1 ± 122 are (Example 13.4)
(13.55)
and all others can be obtained by means of the recurrence relation (true for Bessel
functions in general)
(13.56)
EXAMPLE 13.12Use the recurrence relation (13.56) and the formulas (13.55) to
derive the Bessel functionsJ
± 322(x).
We have, from (13.56) withn 1 = 1122 ,
Jx
x
JxJ
32 12 121
()−+=() 0
−Jx
n
x
Jx J x
nnn+−−+=
112
() () () 0
Jx
x
xJx
x
x
12 1222
()=,sin ()=cos
−ππ
Jx
x
x
n
x
n()∼ sin
2
π 24
ππ
−+
, for large