# Question on Spherical Bessel's equation.

1. Jul 5, 2010

### yungman

Spherical Bessel equation:

$$r^2R''+2rR' + [kr^2-n(n+1)]R = 0 \;\hbox { where }\; k=\lambda^2$$

Boundary condition: $R(a)=0$

Solution :

$$j_n(\lambda_{n,j},r) \;\hbox { where }\; \lambda = \lambda_{n,j}= \frac{\alpha_{n+\frac{1}{2},j}}{a}$$

$$j_n(x)=\sqrt{\frac{\pi}{2x}}J_{n+\frac{1}{2}}(x)$$

I want to find $j_n(\lambda_{n,j}r)$

This is what I have:

$$j_n(\lambda_{n,j}r)=\sqrt{\frac{\pi}{2\lambda_{n,j}r}} \; J_{n+\frac{1}{2}}(\lambda_{n,j}r) = \sqrt{\frac{\pi}{ 2 \frac{\alpha_{(n+\frac{1}{2},j)}}{a}r}} \;\; J_{n+\frac{1}{2}}(\frac{\alpha_{(n+\frac{1}{2},j)}}{a}r)$$

I think I am correct actually the confusion is the order of the Bessel function:

$$p=n+\frac{1}{2} \Rightarrow\; \lambda_p = \lambda_n = \frac{\alpha_{n+\frac{1}{2}}}{a}$$

Can anyone verify this?

Last edited: Jul 5, 2010