Math 381 Fall 2006 Homework # 10
Due: Wednesday, November 29, 2006
I couldn’t find any good problems in the textbook this week so I decided to
write up some of my own.
1. Let zmp and zmq be two different zeros of the Bessel function Jm (z). Show
r
that Jm (zmp ar ) and
R a Jm (zmq a ) are orthogonal with respect to the inner
product hf, gi = 0 f grdr. [Hint: show they are eigenfunctions of the SL
equation
df
m2
d
r
+ λr −
f =0
dr
dr
r
with boundary conditions f (0) finite and f (a) = 0.]
2. Denote by Pn (x) = Pn0 (x) the Legendre polynomials. Recall that Pn (x) is
an eigenfunction to the differential equation
d
2 dg
(1 − x )
+ µn g = 0
dx
dx
where µn = n(n + 1). This is a Sturm-Liouville equation and hence the
R1
eigenfunctions are orthogonal with respect to hf, gi = −1 f gdx. Compute
P2 (x) by using the fact that it’s a polynomial of degree two and orthogonal
to P0 (x) = 1 and P1 (x) = x and normalized so that P2 (1) = 1. Note: in
general you can find Pn (x) in this way if you know P0 , . . . , Pn−1 .
3. Consider again the differential equation
d
dg
(1 − x2 )
+ µg = 0.
dx
dx
Show that if p(x) is a solution which is a polynomial of degree two then
µ = 2 · 3. [Hint: Suppose p(x) = a0 + a1 x + a2 x2 where a2 6= 0 and solve
for the coefficients]. The same method can be generalized to explain why
a degree n polynomial is a solution only if µ = n(n + 1).