Bessel Functions in Python

Summary

The Bessel functions are a testament to the beauty and power of mathematics for describing and understanding our world. This is a very brief introduction to Bessel functions with code on using them in python and a list of references for further reading.

The Bessel Differential Equation

The Bessel differential equation is a classical mathematical equation that has a rich history and a wide range of applications in various fields. It is named after the German mathematician Friedrich Bessel, who introduced it in the early 19th century as a generalization of the circular functions.

The Bessel differential equation has the following form:

where y is the unknown function, x is the independent variable, and n is a constant. The Bessel differential equation arises in various physical and mathematical fields, including atomic physics, thermodynamics, electromagnetism, and statistics.

Bessel Functions of the First Kind

The Bessel differential equation has a family of solutions known as the Bessel functions of the first kind, denoted by J_n(x). These functions are defined by the series expansion:

where $\Gamma$ is the gamma function. The Bessel functions of the first kind have several important properties and applications in mathematics and physics. For example, they are used to describe the radial part of the wave functions for the hydrogen atom, the propagation of electromagnetic waves, and the oscillations of a drumhead.

Bessel Functions of the Second Kind

Another beautiful property of the Bessel differential equation is that it has a second family of solutions known as the Bessel functions of the second kind, which can be expressed concisely in terms of the Bessel functions of the first kind:

where $J_n(x)$ is the Bessel function of the first kind. The Bessel functions of the second kind have similar properties and applications as the Bessel functions of the first kind, but they are defined for a different range of the argument $x$.

Bessel Functions in Python

To plot Bessel functions in Python, you can use the scipy library to compute the Bessel functions and the matplotlib library to generate the plot. Here's an example of how you can plot several Bessel functions of the first kind in Python:

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import jv


# Define the x-values for the plot
x = np.linspace(0, 20, 1000)


# Compute the Bessel functions using list comprehension
y = [jv(n, x) for n in range(4)]


# Plot the Bessel functions
for i, yi in enumerate(y):
    plt.plot(x, yi, label='J{}(x)'.format(i))
plt.xlabel('x')
plt.ylabel('J(x)')
plt.title('Bessel Functions of the First Kind')
plt.legend()
plt.show()        

Note, we used list comprehension to make the code concise.

Replace y in the code block above with y = [yn(n, x) for n in range(10)] to get a plot for Bessel functions of the second kind:

from scipy.special import yn

# Define the x-values for the plot
x = np.linspace(0, 20, 1000)

# Compute the Bessel functions using list comprehension
y = [yn(n, x) for n in range(10)]

# Plot the Bessel functions
for i, yi in enumerate(y):
    plt.plot(x, yi, label='J{}(x)'.format(i))
plt.xlabel('x')
plt.ylabel('J(x)')
plt.title('Bessel Functions of the First Kind')
plt.legend()
plt.show()        

Which gives

Note that I used some additional formatting in my figures. Feel free to add your own styles following the matplotlib.pyplot.plot documentation.

In summary, the Bessel differential equation is a beautiful mathematical equation that has a rich history and a wide range of applications in various fields. It has a family of solutions known as the Bessel functions of the first kind, which have several important properties and applications. It also has a second family of solutions known as the Bessel functions of the second kind, which are defined in terms of the Bessel functions of the first kind. The Bessel differential equation is a testament to the beauty and power of mathematics in describing and understanding our reality.

References of Further Readings

There are many well-known references for Bessel functions, including books, articles, and online resources:

• Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by Milton Abramowitz and Irene A. Stegun (National Bureau of Standards, 1964). This is a classic reference on special functions, including Bessel functions. It provides a comprehensive treatment of the properties and applications of Bessel functions, as well as numerical tables and plots of the functions.
• Bessel Functions, by Frank Bowman (Dover Publications, 1958). This is a comprehensive treatment of Bessel functions, including their properties, applications, and numerical evaluation. It is aimed at a general mathematical audience and does not assume a strong background in advanced mathematics.
• Bessel Functions, by Frank Bowman (Oxford University Press, 1995). This is a more recent edition of the same book, updated to include more recent developments in the theory of Bessel functions.
• "Special Functions", by George E. Andrews, Richard Askey, and Ranjan Roy (Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 1999). This is a comprehensive encyclopedia article on special functions, including Bessel functions. It provides a detailed treatment of the properties and applications of Bessel functions, as well as the connections between Bessel functions and other special functions.
• "Bessel Functions", by Frank Bowman (Scholarpedia, 2009). This is a comprehensive online encyclopedia article on Bessel functions, written by the same author as the book mentioned above. It provides a detailed treatment of the properties and applications of Bessel functions, as well as the connections between Bessel functions and other special functions.
• "Bessel Functions", by Eric Weisstein (Wolfram MathWorld, 2021). This is an online resource on Bessel functions, including