Today, we finish up some loose ends in Chapter 12 and talk about a few exercises.

These are related to half-integer order Bessel functions $J_{n+1/2}(x), Y_{n+1/2}(x)$.

$$ j_n(x) = \sqrt{ \frac{\pi}{2x}} J_{n+1/2}(x) = x^n \left( - \frac{1}{x} \frac{d}{dx} \right)^n \left( \frac{\sin x}{x} \right) $$ $$ y_n(x) = \sqrt{ \frac{\pi}{2x}} Y_{n+1/2}(x) = - x^n \left( - \frac{1}{x} \frac{d}{dx} \right)^n \left( \frac{\cos x}{x} \right) $$

OK. These are analog of 'Rodrigue formula' for the Legendre polynomials, lovely. Unfortunately, we do not have a similar expression for the integer valued Bessel functions $J_n, Y_n$, so I don't know how to derive these guys.

You can read about the first few entries of $j_n$ and $y_n$ on wikipedia

What are they good for? Well, we will see the usual Bessel function is good for solving PDE in cylindrical coordinate in 3D; these will be useful when using spherical coordinate $r, \theta, \phi$.