Bessel Functions

The orthogonal basis set for cylindrical coordinates

$j_l(x)\equiv (-x{)}^l{\left(\frac{1}{x}\frac{\mathrm{d}}{\mathrm{d}x}\right)}^l\frac{\sin x}{x}\to \frac{2^{l}l!}{(2l+1)!}{x}^l\text{ for }x\ll 1$

• Used to solve the infinite sphere well

Example Values

Bessel Function	Value
$j_0(x)$	$\frac{\sin x}{x}$
$j_1(x)$	$\frac{\sin x}{x^2}-\frac{\cos x}{x}$
$j_2(x)$	$\left(\frac{3}{x^3}-\frac{1}{x}\right)-\frac{3}{x^2}\cos x$