|𝔻⟩irac's Student: Refresher: Bessel Functions

Let's talk about Bessel functions! First, who was Bessel? Friedrich Bessel was a German astronomer and mathematician whose namesake was given to the solutions used to solve a specific type of differential equation. The differential equation in question is the following:

\begin{equation}x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + (x^2-n^2)y = 0\label{eq:ode}\end{equation}

This is a second-order linear ordinary differential equation and is known as a canonical ODE because its solutions are special types of functions. For those unfamiliar with second-order linear ODEs let me explain ( I often forget so it's helpful for me as well). The "second-order" just means the highest derivative in the equation, in this case, its two. The ordinary is because there is only a single independent variable $x$. Linear indicates that we don't have terms like $y^2\frac{dy}{dx}$, in other words, the dependent variable and its derivatives only appear to the first power and the coefficients are not in terms of the dependent variable. There is also the case that this equation is homogenous because $y$ is involved with every term.

The solutions to the Bessel equation are called Bessel functions. The functions form a basis because they are linearly independent and there are two flavors of functions called the "first-kind" and the "second-kind". The first-kind, usually denoted as $J_n(x)$, has the form:

\begin{equation}J_n(x) = \frac{1}{\pi} \int_0^{\pi} \cos(n*t - x \sin(t)) dt\end{equation}

where $n$ is an integer. This is actually a special case known as Bessel integrals. The solutions of the first kind can be multiplied by a scalar value and remain solutions to the ODE. Given Bessel functions of the first kind are written in terms of sine and cosine they are oscillatory functions. They are linearly dependent and form an orthogonal basis, given that:

$$\int_0^x J_n(x) \cdot J_m(x) dx = 0 \; m \neq n $$

They also have recurrence relations whereby negative values for $n$ are the same as positive values for $n$. The Bessel functions of the first kind are used to obtain solutions in cylindrical coordinate representations because the solutions are the radial part. In spherical coordinates, the radial part is given by the Bessel functions of the second kind. So what do the Bessel functions of the first kind look like? Let me show it with some Julia code that would compute the Bessel function integral of the first kind for integer values of $n$.


"""
	Jₙ(x;n=1,ν=1000)

First kind Bessel numerical (i.e., trapizodial) integrals for
integer values of n.
""" 
function Jₙ(x,n=1;ν=1000)
    ∫f = 0.0
    for i=0:ν
    	t = π/ν * i
        f = cos(n * t - x * sin(t))
        # trapezoidal weights
        w = (i == 0 || i == ν) ? 0.5 : 1.0 
        ∫f += w * f
    end
    return 1/ν * ∫f
end

The code simply calculates the integral form for the first kind of Bessel function, which is possible when $n$ is an integer. I believe this is also a valid function for non-integer values of $n$ but only when $x$ is real and $\gt 0$. When we plot this function for different values of $n$ we get the following solutions:

Bessel functions (integrals) of the first kind where $n$ is an integer.

So what about Bessel functions of the second kind? When $n$ is not an integer these take the form:

\begin{equation}Y_m(x) = J_m(x) * \cos(m\pi) - \frac{J_{-m}(x)}{\sin(m\pi)}\label{eq:bessel1}\end{equation}

with $m$ being a non-integer value. As you can see, $Y_m(x)$ can be defined in terms of Bessel functions of the first kind, $J_m(x)$ and $J_{-m}(x)$. The functions in eq. (\ref{eq:bessel1}) are clear valid because they are linear combinations of first-kind solutions. One thing of importance is that Bessel functions of the second kind have a singularity at $x=0$ for all values of $m$. Also, keep in mind that $J_m(x)$ is not the function $J_n(x)$ since $m$ is a non-integer. If $m\to +n$, i.e., is an integer and non-negative, then the solution [1] is given by:

\begin{multline}Y_n(x) =-\frac{\left(\frac{x}{2}\right)^{-n}}{\pi}\sum_{k=0}^{n-1} \frac{(n-k-1)!}{k!}\left(\frac{x^2}{4}\right)^k +\frac{2}{\pi} J_n(x) \ln \frac{x}{2} \\ -\frac{\left(\frac{x}{2}\right)^n}{\pi}\sum_{k=0}^\infty (\psi(k+1)+\psi(n+k+1)) \frac{\left(-\frac{x^2}{4}\right)^k}{k!(n+k)!}\label{eq:bessel2}\end{multline}

where $\psi(x)$ is the logarithmic derivative of the gamma function. You can review the Wikipedia entry for Bessel functions which provides more details on the derivation of this solution. This solutions $Y_n(x)$ are linearly independent from $J_n(x)$ hence why we have "second-kind" solutions Bessels equations. There is also an integral form that exists when $x$ is positive and real-valued. So what does the plot look like for eq. (\ref{eq:bessel2})? Below shows the plot for the limiting case discussed.

Bessel functions of the second kind where $x>0$ and $n$ is an integer.

If you want to look at the Julia implementation for the Bessel functions of the second kind you can take a look at the Pluto.jl notebook here. You'll need to set up Julia+Pluto.jl to run the notebook locally. You can also use the "run in cloud" but its not worth it. For Windows users who just want to run the notebook without any setup, check out the PlutoDesktop exe. For python users you can probably just copy the raw notebook file content into an LLM and ask for it to rewrite the code in Python.

In general, the use of Bessel functions of the first kind vs the second kind often depends on the specific problem at hand and the boundary conditions involved. Both types of Bessel functions are solutions to eq. (\ref{eq:ode}), but they behave differently and hence have different applications. Here's a quick rundown:

First Kind ($J_n$): These functions are finite at the origin ($x=0$), and this makes them suitable for problems with boundary conditions that require the solution to be finite at the origin. For example, they often appear in problems of wave propagation and heat conduction in cylindrical or spherical symmetry where the solution must be finite at $r=0$.
Second Kind ($Y_n$): These functions, on the other hand, have a singularity at the origin. They are often useful in problems where the physical system has a singularity at the origin or where the boundary conditions do not require the solution to be finite at the origin. They are also used when the full set of linearly independent solutions to eq. \ref{eq:ode} are needed.

The complete solution to eq. $\ref{eq:ode}$ is a linear combination of Bessel functions, where the coefficients (i.e., amplitudes) are determined by the boundary conditions. In the context of partial differential equations such as Laplace's equation in cylindrical or spherical coordinates, the Bessel function provides solutions to the radial part whereas the spherical harmonics represent the angular parts.

References

[1] Y. L. Luke, Integrals of Bessel Functions (Courier Corporation, 2014).

[2] DLMF: Chapter 10 Bessel Functions, https://dlmf.nist.gov/10.