# Homework Help: Bessel functions

1. Apr 26, 2010

### owlpride

1. The problem statement, all variables and given/known data

Find the expansion of $1 - x^2$ on the interval $0 < x < 1$ in terms of the Eigenfunctions
$$J_0 ( \sqrt{ \lambda_k ^{(0)}} x)$$

(where $\lambda_k ^{(0)}$ denotes the kth root > 0 of $J_0$) of

$$(x u')' + \lambda x u = 0$$
$$u(1) = 0$$
u and u' bounded.

2. Relevant equations

Hint from the text: Use integration by parts and the following identity:
$$\frac{d}{dt} [t^m J_m (t)] = t^m J_{m-1} (t)$$

3. The attempt at a solution

Having never worked with Bessel functions before, I am a bit at a loss for what to do...