Bessel Functions

Bessel functions $J_n(x)$ are solutions to Bessel's differential equation:

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

They arise in problems with cylindrical symmetry: heat conduction in cylinders, electromagnetic waveguides, and quantum mechanics in cylindrical potentials.

Series Definition

The Bessel function of the first kind of order $n$ is defined by the convergent series:

$J_n(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m!\, \Gamma(m+n+1)} \left(\frac{x}{2}\right)^{2m+n}$

For integer $n$, $\Gamma(m+n+1) = (m+n)!$, so the series simplifies.

The first few values:

• $J_0(0) = 1$, $J_0(2.4048) \approx 0$ (first zero)
• $J_1(0) = 0$, $J_1(1) \approx 0.4401$

Recurrence Relation

Given $J_0$ and $J_1$, higher orders are computed by the forward recurrence:

$J_{n+1}(x) = \frac{2n}{x} J_n(x) - J_{n-1}(x)$

This recurrence is numerically stable for computing $J_n$ when going upward in $n$ at fixed $x > 0$.

Important Zeros

These zeros determine the resonant frequencies in cylindrical cavities.

Wronskian

The Wronskian of $J_0$ and $J_1$ satisfies:

$J_0(x) J_1'(x) - J_1(x) J_0'(x) = -\frac{1}{x}$

This identity is a consequence of Bessel's equation and can be used to check numerical accuracy.

Your Task

Implement the three Bessel function routines. Use 20-term series for $J_0$ and $J_1$. For $\Gamma(m+n+1)$ with integer $n$, you may use the factorial $(m+n)!$. Use only Python's math module.