Problem set 7

1. Properties of Bessel Functions

Consider Bessel's differential equation \[ x^2 y'' + xy' + (x^2 - n^2)y = 0, \]

for some integer $n \geq 0$. From a previous problem set, we know that a solution of this equation is given by the Bessel function of thr first kind, $J_n$, which has the series expansion \[ J_n(x) = \left(\frac{x}{2}\right)^n \sum_{k=0}^\infty \frac{(-x^2/4)^k}{k! (k+n)!}. \]

a) Using the series expansion, show that the following recursion relation is true: \[ J_{n+1}(x) = \frac{2n}{x} J_n(x) - J_{n-1}(x). \]

b) For any integer $n \geq 0$ show that \[ \int_0^1 x [J_n(\alpha x)]^2 \, \de{x} = \frac{1}{2} [J_n'(\alpha)]^2. \]

Spoiler

2. Bessel Functions in a Sturm Liouville Problem

a) Determine the bounded eigenfunctions and eigenvalues of the singular Sturm-Liouville problem on $0 \leq x \leq 1$, \[ \frac{d}{dx} \left( x \frac{dy}{dx}\right) + \lambda xy = 0, \qquad y(1) = 0. \]

b) Use a) to obtain the eigenfunction expansion for the bounded solution of the inhomogeneous problem $0 \leq x \leq 1$, \[ \frac{d}{dx} \left(x \frac{dy}{dx}\right) = x, \qquad y(1) = 0. \]

Spoiler

3. Legendre functions

Author's note: This is a really nasty problem (particularly part c and d); why is it even in the course? Full of integration and other algebra-related tricks. It is better that this question is swapped out for something that would be less tedious and more educational.

Consider Legendre's equation \[ (1 - x^2)y'' - 2xy' + \left[ l(l+1) - \frac{m^2}{1-x^2}\right] y = 0. \]

and the self-adjoint form \[ \left[(1 - x^2)y'\right]' + \left[ l(l+1) - \frac{m^2}{1-x^2}\right] y = 0. \]

for integer values $0 \leq m \leq l$.

a) Show that Rodrigue's formula, \[ P_l(x) = \frac{1}{2^l l!} \frac{d^l}{dx^l} \left[(x^2 - 1)^l\right], \]

is a solution to Legendre's equation with $m = 0$ and $l \geq 0$ and satisfies $P_l(1) = 1$.

Spoiler

b) Show that the associated Legendre functions \[ P_l^m(x) = (-1)^m (1-x^2)^{m/2} \frac{d^m[P_l(x)]}{dx^m} \]

are solutions of Legendre's equation for $0 \leq m \leq l$.

Spoiler

c) Show that the following relation is true for $0 \leq m \leq l$: \[ \int_{-1}^1 P_n^m(x) P_l^m(x) \, \de{x} = \begin{cases} 0 & \text{if } l \neq n \\ \frac{2}{2n+1} \frac{(n+m)!}{(n-m)!} & \text{if } l = n. \end{cases} \]

Spoiler