Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
If y 1 and y 2 are eigenfunctions corresponding to the two di€erent eigenvalues
1 ; 2 , respectively,
d
dx r…x† dy
1
‡‰q…x†‡
dx
1 p…x†Šy 1 ˆ 0; a x b; …7:106†
k 1 y 1 …a†‡k 2 y 0 1…a† ˆ0; l 1 y 1 …b†‡l 2 y 0 1…b† ˆ0; …7:106a†
d
dx r…x† dy
2
‡‰q…x†‡
dx
2 p…x†Šy 2 ˆ 0; a x b; …7:107†
k 1 y 2 …a†‡k 2 y 0 2…a† ˆ0; l 1 y 2 …b†‡l 2 y 0 2…b† ˆ0: …7:107a†
Multiplying (7.106) by y 2 and (7.107) by y 1 , then subtracting, we obtain
d
dx r…x†…y 1y2 0 y 2 y1†
0
ˆ… †p…x†y1 y 2 :
Integrating from a to b, and using (7.106a) and (7.107a), we obtain
… 1 2 †
Z b
a
p…x†y 1 y 2 dx ˆ r…x†…y 1 y 0 2 y 2 y 0 1†j b a ˆ 0:
Since 1 6ˆ 2 we have the required result; that is,
Z b
a
p…x†y 1 y 2 dx ˆ 0:
We can normalize these eigenfunctions to make them an orthonormal set, and
so we can expand a given function in a series of these orthonormal eigenfunctions.
We have shown that Legendre's equation is a Sturm±Liouville equation with
r…x† ˆ1 x; q ˆ 0 and p ˆ 1. Since r ˆ 0 when x ˆ1, no boundary conditions
are needed to form a Sturm±Liouville problem on the interval 1 x 1. The
numbers n ˆ n…n ‡ 1† are eigenvalues with n ˆ 0; 1; 2; 3; .... The corresponding
eigenfunctions are y n ˆ P n …x†. Property 2 tells us that
Z 1
1
For Bessel functions we saw that
P n …x†P m …x†dx ˆ 0 n 6ˆ m:
‰xJ 0
n…x†Š 0 ‡
!
n2
x ‡ 2 x J n …x† ˆ0
is a Sturm±Liouville equation (7.104), with r…x† ˆx; q…x† ˆn 2 =x; p…x† ˆx, and
with the parameter now written as 2 . Typically, we want to solve this equation
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