Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
Now multiplying the second equation by sin k and integrating from 0 to , we
obtain
Z (
1
sin k sin…x sin †d ˆ Jk…x†; if k ˆ 1; 3; 5; ...
:
0; if k ˆ 0; 2; 4; ...
0
Adding these two together we obtain Bessel's integral representation
J n …x† ˆ1
Z
0
cos…n x sin †d; n ˆ positive integer: …7:89†
Recurrence formulas for J n …x†
Bessel functions of the ®rst kind, J n …x†, are the most useful, because they are
bounded near the origin. And there exist some useful recurrence formulas between
Bessel functions of di€erent orders and their derivatives.
…1† J n‡1 …x† ˆ2n
x J n…x†J n1 …x†:
…7:90†
Proof:
obtain
or
Di€erentiating both sides of the generating function with respect to t, we
This can be rewritten as
or
X 1
e x…tt1 †=2 x
2 1 ‡ 1
t 2 ˆ X1
nJ n …x†t n1
nˆ1
x
2 1 ‡ 1 X 1
t 2 J n …x†t n ˆ X1
nJ n …x†t n1 :
nˆ1
nˆ1
x
J
2 n …x†t n ‡ x 2
nˆ1
X 1
x
J
2 n …x†t n ‡ x 2
nˆ1
X 1
nˆ1
X 1
nˆ1
J n …x†t n2 ˆ X1
J n‡2 …x†t n ˆ X1
nˆ1
nˆ1
Equating coecients of t n on both sides, we obtain
x
2 J n…x†‡ x 2 J n‡2…x† ˆ…n ‡ 1†J n …x†:
Replacing n by n 1, we obtain the required result.
…2† xJ 0
n…x† ˆnJ n …x†xJ n‡1 …x†:
332
nJ n …x†t n1
…n ‡ 1†J n‡1 …x†t n :
…7:91†