Bessel functions

Bessel functions of the first kind, $J_{α} (x)$ , and of the second kind, $Y_{α} (x)$ , are solutions to the Bessel differential equation:

x^{2} y^{″} + x y^{'} + (x^{2} - α^{2}) y = 0

Here, $α$ can be any real or complex number, and $x$ is the independent variable.

Domain of $α$ :
$α$ can be any real or complex number. The value of $α$ determines the order of the Bessel function. For integer orders, the functions have particularly simple representations, but they're well-defined for non-integer orders as well.
Domain of $x$ :
Bessel functions are defined for all real numbers $x$ . Depending on the context:
- For positive $x$ , the Bessel functions oscillate and are well-defined.
- For $x = 0$ :
  - $J_{0} (0) = 1$ and $J_{α} (0) = 0$ for $α \neq 0$ .
  - $Y_{α} (0)$ is not defined (it approaches negative infinity).
- For negative $x$ :
  The Bessel functions can be expressed in terms of their values for positive arguments using certain relations. For example, for integer order $n$ :
  $J_{- n} (x) = (- 1)^{n} J_{n} (x)$ $Y_{- n} (x) = (- 1)^{n} Y_{n} (x)$
  However, for non-integer $α$ , the functions $J_{α} (x)$ and $J_{- α} (x)$ are linearly independent.

So, in summary:

$α$ (order) can be any real or complex number.
$x$ (argument) can be any real number, but one has to be careful at $x = 0$ especially for the Bessel function of the second kind.

Properties:

\begin{aligned} J_{ν} J_{- ν + 1} + J_{- ν} J_{ν - 1} & = \frac{2 \sin ν π}{π x}, \\ J_{ν} J_{- ν - 1} + J_{- ν} J_{ν + 1} & = - \frac{2 \sin ν π}{π x}, \\ J_{ν} Y_{ν}^{'} - J_{ν}^{'} Y_{ν} & = \frac{2}{π x}, \\ J_{ν} Y_{ν + 1} - J_{ν + 1} Y_{ν} & = - \frac{2}{π x} \end{aligned}