Mathematical Methods for Physicists: A concise introduction - Site Map

BESSEL'S EQUATION
Now a 2m is the coecient of x ‡2m in the series (7.72) for y. Hence it would be
convenient if a 2m contained the factor 2 ‡2m in its denominator instead of just 2 2m .
To achieve this, we write
a 2m ˆ
…1† m
2 ‡2m m!… ‡ m†… ‡ 2†… ‡ 1† …2 a 0 †:
Furthermore, the factors
… ‡ m†… ‡ 2†… ‡ 1†
suggest a factorial. In fact, if were an integer, a factorial could be created by
multiplying numerator by !. However, since is not necessarily an integer, we
must use not ! but its generalization … ‡ 1† for this purpose. Then, except for
the values
ˆ1; 2; 3; ...
for which … ‡ 1† is not de®ned, we can write
a 2m ˆ
…1† m
2 ‡2m m!… ‡ m†… ‡ 2†… ‡ 1†… ‡ 1† ‰2 … ‡ 1†a 0 Š:
Since the gamma function satis®es the recurrence relation z…z† ˆ…z ‡ 1†, the
expression for a 2m becomes ®nally
a 2m ˆ
…1† m
2 ‡2m m!… ‡ m ‡ 1† ‰2 … ‡ 1†a 0 Š:
Since a 0 is arbitrary, and since we are looking only for particular solutions, we
choose
a 0 ˆ
1
2 … ‡ 1† ;
so that
a 2m ˆ
…1† m
2 ‡2m m!… ‡ m ‡ 1† ; a 2m‡1 ˆ 0
and the series for y is, from Eq. (7.72),
" #
y…x† ˆx 1
2 … ‡ 1† x 2
2 ‡2 … ‡ 2† ‡ x 4
2 ‡4 2!… ‡ 3† ‡
ˆ X1
mˆ0
…1† m
2 ‡2m m!… ‡ m ‡ 1† x‡2m : …7:76†
The function de®ned by this in®nite series is known as the Bessel function of the
®rst kind of order and is denoted by the symbol J …x†. Since Bessel's equation
323