Mathematical Methods for Physicists: A concise introduction - Site Map

BESSEL'S EQUATION
Proof:
J n …x† ˆX1
kˆ0
Di€erentiating both sides once, we obtain
from which we have
J 0
n…x† ˆX1
kˆ0
xJ 0
n…x† ˆnJ n …x†‡x X1
…1† k
k!…n ‡ k ‡ 1†2 n‡2k xn‡2k :
…n ‡ 2k†…1† k
k!…n ‡ k ‡ 1†2 n‡2k xn‡2k1 ;
kˆ1
…1† k
…k 1†!…n ‡ k ‡ 1†2 n‡2k1 xn‡2k1 :
Letting k ˆ m ‡ 1 in the sum on the right hand side, we obtain
xJ 0
n…x† ˆnJ n …x†x X1
ˆ nJ n …x†xJ n‡1 …x†:
mˆ0
…1† m
xn‡2m‡1
m!…n ‡ m ‡ 2†2n‡2m‡1 …3† xJ 0
n…x† ˆnJ n …x†‡xJ n1 …x†:
…7:92†
Proof:
Di€erentiating both sides of the following equation with respect to x
x n J n …x† ˆX1
kˆ0
…1† k
k!…n ‡ k ‡ 1†2 n‡2k x2n‡2k ;
we have
d X 1
dx
kˆ0
d
dx fxn J n …x†g ˆ x n J 0
n…x†‡nx n1 J n …x†;
…1† k x 2n‡2k
2 n‡2k k!…n ‡ k ‡ 1† ˆ X1
Equating these two results, we have
kˆ0
ˆ x n X1
…1† k x 2n‡2k1
2 n‡2k1 k!…n ‡ k†
kˆ0
ˆ x n J n1 …x†:
x n J 0
n…x†‡nx n1 J n …x† ˆx n J n1 …x†:
333
…1† k x …n1†‡2k
2 …n1†‡2k k!‰…n 1†‡k ‡ 1Š