Bessel functions

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

$$ x^2 y'' + x y' + (x^2 - \alpha^2) y = 0 $$

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

1. Domain of $\alpha$:

$\alpha$ can be any real or complex number. The value of $\alpha$ 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.

2. 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 $J_\alpha(0) = 0$ for $\alpha \neq 0$.

- $Y_\alpha(0)$ 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 $\alpha$, the functions $J_{\alpha}(x)$ and $J_{-\alpha}(x)$ are linearly independent.

So, in summary:

$\alpha$ (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:

$$ \begin{aligned} J_{\nu} J_{-\nu+1}+J_{-\nu} J_{\nu-1} &=\frac{2 \sin \nu \pi}{\pi x}, \\ J_{\nu} J_{-\nu-1}+J_{-\nu} J_{\nu+1} &=-\frac{2 \sin \nu \pi}{\pi x}, \\ J_{\nu} Y_{\nu}^{\prime}-J_{\nu}^{\prime} Y_{\nu} &=\frac{2}{\pi x}, \\ J_{\nu} Y_{\nu+1}-J_{\nu+1} Y_{\nu} &=-\frac{2}{\pi x} \end{aligned} $$

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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