Tru Physics

Bessel Functions

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Introduction

Bessel functions, named after Friedrich Bessel, are canonical solutions to Bessel’s differential equation:

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

This equation arises in many physical problems, including heat conduction, vibrations of circular membranes, and quantum mechanics.

Bessel Functions of the First Kind

The Bessel function of the first kind J_n(x) is defined by the series:

J_n(x) = \dfrac{1}{\pi} \displaystyle\int_0^\pi \cos(n t - x \sin t) dt

Alternatively, it can be expressed as a power series:

J_n(x) = \dfrac{1}{\pi} \displaystyle\sum_{m=0}^\infty \dfrac{(-1)^m}{m!(m+n)!}\left(\dfrac{x}{2}\right)^{2m+n}

The plot below gives a better understanding of that this summation results in (black). It’s optimized for desktop mode, so switch over and open in a new tab to explore further.

Bessel Functions of the Second Kind

The Bessel function of the second kind, also known as the Neumann function and denoted by Y_n(x), is defined as:

Y_n(x) = \dfrac{J_n(x) \cos(n \pi) - J_{-n}(x)}{\sin(n \pi)}

This function is singular at the origin and often arises in problems with cylindrical symmetry.

Modified Bessel Functions

When dealing with problems involving cylindrical symmetry in two dimensions, we often encounter modified Bessel functions. They are solutions to the modified Bessel’s equation and are denoted as I_n(x) and K_n(x).

Orthogonality and Normalization

Bessel functions of the first kind of different orders are orthogonal over the interval [0, \infty) with respect to the weight function x. The orthogonality relationship is given by:

\displaystyle\int_0^\infty J_m(x)J_n(x)x dx = 0

for m \neq n and

\displaystyle\int_0^\infty [J_n(x)]^2 x dx = \dfrac{1}{2}

for m = n.

Conclusion

Bessel functions are a rich and powerful class of functions with a broad range of applications in physics and engineering. From the diffraction of waves to the modes of a vibrating drum, their unique properties provide solutions to a wide variety of problems.

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