====== 2020-03-11, Wednesday ======
Today we talked about section 13-15, the second Bessel function, an aside on Gamma function, 
the shape of Bessel functions, and some recursion relations. 

The explanation from Boas are quite detailed, so I won't repeat it here. A rough outline of the lecture is the following
  - Recall the Bessel equation. 
  - The formula for $J_p(x)$.
  - For $p>0$, the two cases of $J_{-p}(x)$. $p$ is an integer, or $p$ is a non-integer. 
  - For $p$ a non-integer, $J_{-p}(x)$ is linearly independent of $J_p(x)$, since they have different leading order term, one is $x^p$ the other is $x^{-p}$. 
  - For $p$ an integer,  $$ J_{-p}(x) = (-1)^p J_p(x),$$ we get the second differential equation by taking the limit $$ N_p(x) = \lim_{q \to p} \frac{\cos(q \pi) J_q(x) - J_{-q}(x) }{ \sin(q \pi) } $$ Then $N_p(x)$ is well-define even for $p$ an integer. 

The shape of $J_p(x)$ is an oscillation damping. Try [[https://www.wolframalpha.com/]], with input 
<code>
BesselJ[0,x]
</code> 
This will tell you something about $J_0(x)$. Change 0 to other number and play with it. The Neumann function $Y_n(x)$ is called 'BesselY[n,x]'. The program will show the real and imaginary part for $x < 0$. For us, one only need to look at $x>0$'s real part. 

For recursion relation, we checked the equation (15.2). Basically, one just plug in the general fomula of $J_p(x)$ and simplify. 


===== Exercises =====
  * 13.3, 13.4
  * 15.3, 15.7, 15.8