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 $r, \theta, \phi$.

$H_n^1(x), H^2_n(x)$ to $J_n(x), Y_n(x)$ are like $e^{ix}$ and $e^{-ix}$ to $\sin x, \cos x$. They are complex valued functions.

In real life, I have encountered them when solving Dirac equation on expanding universe.

The function is named after a German mathematician Hermann Hankel. He is also known for 'Hankel contour',some contour integral expression for $J_n$ and $H_n$

The $I_p(x)$ and $K_p(x)$ are related to Bessel function when you replace $x$ by $ix$ in the input.

Just convenient names.

This function is pretty popular and useful. It is worth studying this in more details $Ai(x)$.

It solves equation of the type $$ (d/dx)^2 y(x) - x y(x) = 0. $$ Its solution has the property that, it is osillatory for $x < 0$ and have exponential decay for $x > 0$, indeed, the oscillation freqency is $\omega = \sqrt{-x}$, if you compare this with Harmonic oscillator $$ (d/dx)^2 y(x) + \omega^2 y(x) = 0$$ The solution to which is $e^{\pm i \omega x}$ and we know imaginary $\omega$ means exponetial dampling or growth.

The Airy function is used to model transition behavior in quantum mechanics, when you go from the 'allowed region' (total energy > potential energy) to 'forbidden region' (other wise).

We can see the asymptotic behavior of $Ai(x)$ for $x \to -\infty$ and $x \to +\infty$,