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

1. Recall the Bessel equation.
2. The formula for $J_p(x)$.
3. For $p>0$, the two cases of $J_{-p}(x)$. $p$ is an integer, or $p$ is a non-integer.
4. 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 $x^{-p}$.
5. 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

BesselJ[0,x]

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.

• 13.3, 13.4
• 15.3, 15.7, 15.8