Mathematical Methods for Physicists: A concise introduction - Site Map

BESSEL'S EQUATION
Solution:
Taking derivative of the quantity xJ 1 …x† with respect to x, we obtain
d
dx fxJ 1…x†g ˆ J 1 …x†‡xJ 0
1…x†:
Then using Eq. (7.92) with n ˆ 1, xJ 0
1…x† ˆJ 1 …x†‡xJ 0 …x†, we ®nd
thus,
d
dx fxJ 1…x†g ˆ J 1 …x†‡xJ 0
1…x† ˆxJ 0 …x†;
Z t
0
xJ 0 …x†dx ˆ xJ 1 …x†j t 0 ˆ tJ 1…t†:
Approximations to the Bessel functions
For very large or very small values of x we might be able to make some approximations
to the Bessel functions of the ®rst kind J n …x†. By a rough argument, we
can see that the Bessel functions behave something like a damped cosine function
when the value of x is very large. To see this, let us go back to Bessel's equation
(7.71)
x 2 y 00 ‡ xy 0 ‡…x 2 2 †y ˆ 0
and rewrite it as
y 00 ‡ 1 x y 0 ‡
1 2
x 2 !
y ˆ 0:
If x is very large, let us drop the term 2 =x 2 and then the di€erential equation
reduces to
y 00 ‡ 1 x y 0 ‡ y ˆ 0:
Let u ˆ yx 1=2 , then u 0 ˆ y 0 x 1=2 ‡ 1 2 x1=2 y, and u 00 ˆ y 00 x 1=2 ‡ x 1=2 y 0 1 4 x3=2 y.
From u 00 we have
Adding y on both sides, we obtain
y 00 ‡ 1 x y 0 ˆ x 1=2 u 00 ‡ 1
4x 2 y:
y 00 ‡ 1 x y 0 ‡ y ˆ 0 ˆ x 1=2 u 00 ‡ 1
4x 2 y ‡ y;
x 1=2 u 00 ‡ 1
4x 2 y ‡ y ˆ 0
335