Spherical Bessel functions

Remarks

They solve $x^2 y'' + 2x y' + \left( x^2 - n(n+1) \right) y = 0$, which is the radial equation of $\Delta u + u = 0$ in spherical coordinates. Example: $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$, so $j_0(x) = \frac{\sin x}{x}$; one checks $x^2 j_0'' + 2x j_0' + x^2 j_0 = x (x j_0)'' + x^2 j_0 = x (\sin x)'' + x \sin x = 0$. Likewise $y_0(x) = -\frac{\cos x}{x}$. In general $j_n(x) = (-x)^n \left( \frac{1}{x} \frac{d}{dx} \right)^n \frac{\sin x}{x}$. The spherical Hankel functions are $h^{(1)}_n = j_n + i y_n$ and $h^{(2)}_n = j_n - i y_n$; for instance $h^{(1)}_0(x) = -i \frac{e^{ix}}{x}$. A non-example: $J_n(x)$ for an integer $n$ is a cylindrical, not a spherical, Bessel function; it is not elementary.