Mathematical Methods for Physicists: A concise introduction - Site Map

BESSEL'S EQUATION
Similarly, the coecient of the term t n which is made up of those terms for
which k m ˆ n is just J n …x†:
X 1
kˆ0
…1† k
…k ‡ n†!k!2 2k‡n x2k‡n ˆ J n …x†:
This shows clearly that the coecients in the Laurent expansion (7.88) of the
generating function are just the Bessel functions of integral order. Thus we
have proved Eq. (7.87).
Bessel's integral representation
With the help of the generating function, we can express J n …x† in terms of a
de®nite integral with a parameter. To do this, let t ˆ e i in the generating function,
then
e x…tt1 †=2 ˆ e x…ei e i †=2 ix sin
ˆ e
Substituting this into Eq. (7.87) we obtain
cos…x sin †‡i sin…x cos † ˆ X1
ˆ cos…x sin †‡i sin…x cos †:
nˆ1
ˆ X1
1
J n …x†…cos ‡ i sin † n
J n …x† cos n ‡ i X1 J n …x† sin n:
1
Since J n …x† ˆ…1† n J n …x†; cos n ˆ cos…n†, and sin n ˆsin…n†, we have,
upon equating the real and imaginary parts of the above equation,
cos…x sin † ˆJ 0 …x†‡2 X1
sin…x sin † ˆ2 X1
nˆ1
nˆ1
J 2n …x† cos 2n;
J 2n1 …x† sin…2n 1†:
It is interesting to note that these are the Fourier cosine and sine series of
cos…x sin † and sin…x sin †. Multiplying the ®rst equation by cos k and integrating
from 0 to , we obtain
1
Z
0
(
cos k cos…x sin †d ˆ Jk…x†; if k ˆ 0; 2; 4; ...
:
0; if k ˆ 1; 3; 5; ...
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