Bessel Functions

Links:

The Bessel functions are canonical solutions $y (x)$ of Bessel's differential equation: $$x^2 y'' + xy' (x^2-\alpha^2)y = 0$$
for arbitrary complex number $α$ , which represents the order of the Bessel function. We see that $α$ and $- α$ produce the same differential equation, it is conventional to define different Bessel functions for these two values.

Bessel's equation arises when finding solutions to Laplace's equation and the Helmhotz equation in cylindrical or spherical coordinates. In solving problems in cylindrical systems, one obtains Bessel functions of integer order $(α = n)$ ; in spherical coordinates, one obtains half-integer orders $(α = n + 1 / 2)$

Bessel functions of the First Kind

Calculating the Bessel Functions From the ODE, we get that $$J_\alpha(x) = \sum_{m = 0}^\infty \frac{(-1)^m}{m! \Gamma(m+\alpha +1)}\left(\frac{x}{2}\right)^{2m+\alpha}$$This is called the Bessel function of the first kind of order $α$ . We see that the the Gamma Function makes an appearance. The Bessel function of the first kind is an entire function if $α$ is an integers, otherwise it is a multi-valued function with a singularity at $0$ .

For non-integer $α$ , the functions $J_{α}$ and $J_{- α}$ are linearly independent, and are two solutions of the differential equation. For integer order $n$ , the following relationship is valid: $$J_{-n}(x) = (-1)^nJ_n(x)$$
Another definition of the Bessel function, for integer values, is possible using an integral representation: $$J_n(x) = \frac1{2\pi}\int_{-\pi}^\pi e^{ix\sin\theta}e^{-in\theta}, d\theta = \frac1{\pi} \int_0^\pi \cos(n\theta - x \sin\theta), d\theta$$
This integral representation can be interpreted as, the $n$ th Fourier coefficient of the function $e^{i x \sin θ}$ . So $$e^{ix\sin \theta} = \sum_{n = -\infty}^\infty J_n(x) e^{in\theta}$$
We have a couple of properties, for $n$ th order Bessel functions of the first kind, for $n \in Z$ :

$J_{n} (x)$ is real for all real $x$
$J_{- n} (x) = J_{n} (- x) = (- 1)^{n} J_{n} (x)$
$2 J_{n}^{'} (x) = J_{n - 1} (x) - J_{n + 1} (x)$
$(\frac{2 n}{x}) J_{n} (x) = J_{n - 1} (x) + J_{n + 1} (x)$
$(x^{- n} J_{n} (x))^{'} = - x^{- n} J_{n + 1} (x)$
$(x^{n} J_{n} (x))^{'} = x^{n} J_{n - 1} (x)$
For all integers $n$ and all real numbers $a$ and $b$ we have that $$J_n(a+b) = \sum_{k \in \Bbb Z} J_k(a) J_{n-k}(b)$$
Another formula for $J_{n} (x)$ that allows one to define Bessel functions for non-integer valued of $n$ , $n > - 1 / 2$ is $$J_n(x) = \frac{(x/2)^n}{\Gamma(n+1/2)\sqrt \pi} \int_{-1}^1 e^{ixt} (1-t^2)^{n-1/2}, dt$$
We have the special case, that $$J_{1/2}(x) = \sqrt{\frac{2}{\pi}} x^{-1/2}\sin x$$
We can calculate