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Unformatted text preview: ation (1.17). Let us focus now on this
important case. If we set
1
a0 = p ,
2 p!
we obtain
a2 = −a0
1
1
=−
= (−1)
4p + 4
4(p + 1) 2p p! a4 =
= 1
−a2
=
8p + 16
8(p + 2)
1
2(p + 2) 1
2 p+4 p+2 1
2 1
2 p+2 1
=
1!(p + 1)! 1
= (−1)2
1!(p + 1)! 1
2 and so forth. The general term is
a2m = (−1)m 1
2 p+2m 24 1
,
1!(p + 1)! 1
.
m!(p + m)! p+4 1
,
2!(p + 2)! 1
0.8
0.6
0.4
0.2
2 4 6 8 10 12 14 0.2
0.4
Figure 1.1: Graph of the Bessel function J0 (x).
Thus we ﬁnally obtain the power series solution
y= x
2 p∞
m=0 (−1)m 1
m!(p + m)! x
2 2m . The function deﬁned by the power series on the righthand side is called the
pth Bessel function of the ﬁrst kind , and is denoted by the symbol Jp (x). For
example,
∞
x 2m
1
J0 (x) =
(−1)m
.
(m!)2 2
m=0
Using the comparison and ratio tests, we can show that the power series expansion for Jp (x) has inﬁnite radius of convergence. Thus when p is an integer,
Bessel’s equation has a nonzero solution which is real analytic at x = 0.
Bessel functions are so important that Mathematica includes them in its
library of builtin fu...
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 Winter '14
 Equations

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