Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
or
the solution of which is
u 00 ‡ 1
4x 2 ‡ 1 x 1=2 y ˆ u 00 ‡ 1
4x 2 ‡ 1 u ˆ 0;
u ˆ A cos x ‡ B sin x:
Thus the approximate solution to Bessel's equation for very large values of x is
y ˆ x 1=2 …A cos x ‡ B sin x† ˆCx 1=2 cos…x ‡ †:
A more rigorous argument leads to the following asymptotic formula
J n …x† 2 1=2
cos x x
4 n
: …7:94†
2
For very small values of x (that is, near 0), by examining the solution itself and
dropping all terms after the ®rst, we ®nd
J n …x†
x n
2 n …n ‡ 1† : …7:95†
Orthogonality of Bessel functions
Bessel functions enjoy a property which is called orthogonality and is of general
importance in mathematical physics. If and are two di€erent constants, we
can show that under certain conditions
Z 1
0
xJ n …x†J n …x†dx ˆ 0:
Let us see what these conditions are. First, we can show that
Z 1
0
xJ n …x†J n …x†dx ˆ J n…†J 0
n…†J n …†J 0
n…†
2 2 : …7:96†
To show this, let us go back to Bessel's equation (7.71) and change the independent
variable to x, where is a constant, then the resulting equation is
x 2 y 00 ‡ xy 0 ‡… 2 x 2 n 2 †y ˆ 0
and its general solution is J n …x†. Now suppose we have two such equations, one
for y 1 with constant , and one for y 2 with constant :
x 2 y 00
1 ‡ xy 0 1 ‡… 2 x 2 n 2 †y 1 ˆ 0; x 2 y 00
2 ‡ xy 0 2 ‡… 2 x 2 n 2 †y 2 ˆ 0:
Now multiplying the ®rst equation by y 2 , the second by y 1 and subtracting, we get
x 2 ‰y 2 y 00
1 y 1 y 00
2 Š‡x‰y 2 y 0 1 y 1 y 0 2Šˆ… 2 2 †x 2 y 1 y 2 :
336