Bessel Function

Generally, one of the solution of the ODE, Bessel Differential Equation, $$ x^2y'' + xy' + (x2-\nu2)y = 0 $$

is of the form $$ y = x^\nu \sum_{m=0}^\infty \frac{(-1)^m x{2m}}{2{2m+\nu}m!\Gamma(m+\nu+1)} $$

which is denoted by $ y = cJ_\nu(x) $, with $c$ constant.

This fact can be shown by injecting Frobenius series with $r=\pm \nu$.

This is a second order ODE, which two linearly-dependent solutions are required for a general solution. In most case, it can be proven with Abel's Identity that $y^* = J_{-\nu}(x)$ is eligible for the other solution. For some cases, however, $J_{-\nu}$ fails to be linearly-dependent with $J_\nu$. Reduction of order is then applied to get a new function, $Y_\nu$, to be the other solution.

$J_\nu$ is called Bessel function of the first kind of order $\nu$, while $Y_\nu$ is called Bessel function of the second kind of order $\nu$. They turn out to be a set of (orthogonal) basis in the cylindrical coordination, which is similar to $\sin$ and $\cos$ in Cartesian coordination. Many important relationships arise from this property, giving us insight of such systems.

Properties

1. $J_{-\nu}(x) = (-1)^\nu J_\nu(x), \nu \in \mathbb{N}$. This tells why they are linearly dependent.
2. $\frac{d}{dx} [x^\nu J_\nu(x)] = x^\nu J_{\nu-1} (x)$.
3. ...

Written with StackEdit.