F2 - bessel functions

image: Aspire8

x^{2} \frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} + (x^{2} - α^{2}) y = 0

$α$ is the order of the bessel function that determines the shape of the solution
it can be any complex number
when $α \in Z$ , the solutions are called cylinder functions or cylindrical harmonics as they naturally arise when solving problems in cylindrical coordinates
when $α \in Z / 2$ , the solutions are called spherical bessel functions
since the ODEs are second-order, there will be two linearly independent solutions
they are called the first and the second kind, $J_{α}$ and $Y_{α}$
for non-integer $α$ , the functions $J_{α}$ and $J_{- α}$ are linearly independent, and are therefore the two solutions
for integer order, $n :$

J_{- n} (x) = (- 1)^{n} J_{n} (x)

so the two solutions are no longer linearly independent, so the second linearly independent solution is the bessel function of the second kind

image: Aspire8

the bessel functions of the second kind are also known as weber or neumann functions
they have a singularity at $x = 0$
simialr to the functions of the first kind, when $α \in Z :$

Y_{- n} (x) = (- 1)^{n} Y_{n} (x)