====== 2020-03-30, Monday ======
Today, we finish up some loose ends in Chapter 12 and talk about a few exercises.

===== Other Kinds of Bessel Functions (Boas 12.17)=====

==== Speherical Bessel function $j_n(x), y_n(x)$ ====
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 [[https://en.wikipedia.org/wiki/Bessel_function#Spherical_Bessel_functions:_jn,_yn | 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 $r, \theta, \phi$. 


==== Hankel Function ====
$H_n^1(x), H^2_n(x)$ 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 $H_n$

==== Hyperbolic Bessel Function ====
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. 

==== Airy Function ====
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 $x \to +\infty$, 

===== Other Special Function (Boas) =====