Bessel’s differential equation is given by:

It is a 2-dimensional wave equation, having many applications in physics including the notable examples:

- describing electromagnetic waves in a cylindrical wave-guide
- solving the radial Schrodinger Equation for a free particle
- modes of vibration on a circular drum.

For integer it has the series solution:

as

We can use python to approximate using increasingly large .

The function could look something like this;

```
def bessj(n, x, N):
partial_sum = 0
from math import factorial
for l in range(N+1):
partial_sum += (-1)**l/(2**(2*l+n)*factorial(l)*factorial(n+l))*x**(2*l+n)
return partial_sum
```

We could then plot the bessel function approximation for larger values of . We’ll also plot .

```
import numpy as np
import matplotlib.pyplot as plt
def bessel_plotter(N):
plt.xlabel("x")
plt.ylabel("J_n(x)")
plt.title(f"Bessel Function Types (Order {N})")
x_domain = np.linspace(0, 15, 1000)
for kind in [1, 2, 3, 4, 5, 6]:
plt.plot(x_domain, bessj(kind, x_domain, N), label = f'n = {kind}')
return plt.legend()
```

With abit of matplotlib magic we can animate this process:

Beautiful!