13 - Bessel Functions

Motivation for Bessel's Equation

Arises from separation of variables in Laplacian in Cylindrical Polar Coordinates

Consider vibrating membrane of circular drum

Transverse Displacement $w(x,y,t)$ of membrane at $t$ and position $(x,y)$ satisfies
Two-Dimensional Wave Equation

$$\frac{1}{c^2}\frac{{\partial }^{2}w}{\partial t^2}={\nabla }^{2}w=\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2}$$

where $c$ is a constant

If membrane is pinned at boundary of disk of radius $a$ for then for $x^2+y^2<a^2$

$$w=0\text{ at }{x}^2+y^2=a^2$$

Look for a normal mode where membrane oscillates with frequency $\omega$
Hence displacement is in form

$$w(x,y,t)=u(x,y)\cos (wt+\varphi )$$

By substitution then $u$ satisfies Helmholtz Equation

$${\nabla }^{2}u+\lambda u=0$$

where $\lambda =\frac{w^2}{c^2}$

Switching to plane polar coordinates $(r,\theta )$ such that $(x,y)=r(\cos \theta ,\sin \theta )$
Hence through substitution

$$\begin{array}{rl}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\theta }^2}+\lambda u& =00\le r<a,0\le \theta \le 2\pi \\ u& =0r=a,0\le \theta \le 2\pi \\ u& \text{ is }2\pi -\text{periodic}\text{ in }\theta \end{array}$$

Hence it is a PDE Eigenvalue Problem
As $u\equiv 0$ satisfies equation then need to find $\lambda$ with non-trivial solutions $u(r,\theta )$
Since $u(r,\theta )$ is periodic in $\theta$, expand $u$ into Fourier Series of Form

$$u(r,\theta )=U_0(r)+\sum _{n=1}^{\infty }{U}_n(r)\cos (n\theta )+V_n(r)\sin (n\theta )$$

Substituting Fourier Series into PDE then

$$\begin{array}{rl}\frac{1}{r}(r{U}_n^{\prime }(r){)}^{\prime }+\left(\lambda -\frac{n^2}{r^2}\right)U_n(r)& =0,\text{ for }0\le r<a\\ U_n(r)& =0,\text{ at }r=a\end{array}$$

where the same equation and boundary condition hold for $V_n(r)$

Let $U_n(r)=y(x)$ where $x={\lambda }^{1/2}r$ then

$$x^2{y}^{\prime \prime }(x)+x_6{y}^{\prime }(x)+(x^2-n^2)y(x)=0$$

where it is Bessel’s Equation of Order $n$

Bessel's Equation of Order $n$

$$x^2{y}^{\prime \prime }(x)+x{y}^{\prime }(x)+(x^2-n^2)y(x)=0$$

Note that it arises from separation of variables in Laplacian in Cylindrical Polar Coordinates

Properties of Bessel's Equation

Regular Singular Point at $x=0$ with indicial equation

$$F(\alpha )={\alpha }^2-n^2=0$$

with solutions

$${\alpha }_1=n,{\alpha }_2=-n$$

One solution is given by Frobenius Series about $x=0$ with indicial exponent ${\alpha }_1=n$
Second solution is given by Frobenius Series with exponent $\alpha =-n$ plus $\log (x)$ times first sol

Bessel Functions of First Kind

First Frobenius Series, with specific normalising of the leading coefficient in the expansion

$$J_n(x)={\left(\frac{x}{2}\right)}^n\sum _{k=0}^{\infty }\frac{(-1{)}^k}{k!(k+n)!}{\left(\frac{x}{2}\right)}^{2k}$$

for integer $n\ge 0$

Graph

Bessel Functions of Second Kind

Second Frobenius Series, also with specific normalising of the leading coefficient

$$\begin{array}{r}{Y}_n(x)=\frac{2}{\pi }\log \left(\frac{x}{2}\right)J_n(x)-\frac{1}{\pi }{\left(\frac{2}{x}\right)}^n\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}{\left(\frac{x^2}{4}\right)}^k\\ -\frac{1}{\pi }{\left(\frac{x}{2}\right)}^n\sum _{k=0}^{\infty }\frac{[\psi (k+1)+\psi (n+k+1)]}{k!(n+k)!}{\left(-\frac{x^2}{4}\right)}^k\end{array}$$

where diagamma function $\psi (m)$ for integer $m\ge 1$ is defined by

$$\psi (m)=-\gamma +\sum _{k=1}^{m-1}{k}^{-1}$$

and $\gamma =0.5772\dots$ is the Euler-Mascheroni Constant

Graph

Properties of Bessel Functions

1. Bessel’s Equation only has one singular point for finite $x$ so

$$J_n\text{ and }{Y}_n\text{ have infinite radius of convergence}$$

2. $J_n$ and $Y_n$ are oscillating functions decaying slowly as $x\to \infty$
   Each has an infinite set of discrete zeros in $x>0$

3. As $x\to 0$, behaviours of the two kinds of Bessel Function are quite different
   For first kind then $J_n(0)=0$ if $n>0$ and $J_0(0)=1$
   Whilst second kind Bessel functions are singular with $Y_{n}x)\to -\infty$ as $x\to \infty$

4. Two recursion relations can be derived

$$J_{n+1}(x)=\frac{2n}{x}{J}_n(x)-J_{n-1}(x),J_{n+1}(x)=-2J_n^{\prime }(x)+J_{n-1}(x)$$

Same relations hold for second-kind Bessel functions $Y_n$

Zeroes of Bessel Functions of the First Kind $J_n(x)$

$m$	$j_{0,m}$	$j_{1,m}$	$j_{2,m}$	$j_{3,m}$	$j_{4,m}$
1	2.40483	3.83171	5.13562	6.38016	7.58834
2	5.52008	7.01559	8.41724	9.76102	11.0647
3	8.65373	10.1735	11.6198	13.0152	14.3725
4	11.7915	13.3237	14.796	16.2235	17.616
5	14.9309	16.4706	17.9598	19.4094	20.8269

First five zeros of $J_n$ with $n=0,1,2,3,4$

Zeroes of Bessel Functions of the Second Kind $Y_n(x)$

|$m$|$y_{0,m}$|$y_{1,m}$|$y_{2,m}$|$y_{3,m}$|$y_{4,m}$|
| --- | -------- | ------- | ------- | ------- | ------- |
| 1 | 0.893577 | 2.19714 | 3.38424 | 4.52702 | 5.64515 |
| 2 | 3.95768 | 5.42968 | 6.79381 | 8.09755 | 9.36162 |
| 3 | 7.08605 | 8.59601 | 10.0235 | 11.3965 | 12.7301 |
| 4 | 10.2223 | 11.7492 | 13.21 | 14.6231 | 15.9996 |
| 5 | 13.3611 | 14.8974 | 16.379 | 17.8185 | 19.2244 |

First five zeros of $Y_n$ with $n=0,1,2,3,4$

General Solution of Polar ODE

Let ODE be

$$\begin{array}{rl}\frac{1}{r}(r{U}_n^{\prime }(r){)}^{\prime }+\left(\lambda -\frac{n^2}{r^2}\right)U_n(r)& =0,\text{ for }0\le r<a\\ U_n(r)& =0,\text{ at }r=a\end{array}$$

from the motivation of Bessel’s Equation of Order $n$

Then it has general solution

$$U_n(r)=C_1{J}_n({\lambda }^{1/2}r)+C_2{Y}_n({\lambda }^{1/2}r)$$

for some arbitrary constants $C_1$ and $C_2$

As displacement must be bounded as $r\to 0$, hence $C_2=0$ to remove singularity in $Y_n$
As $U_n(r)$ is non-trivial then $C_1\ne 0$ so boundary conditions leads to

$$J_n({\lambda }^{1/2}a)=0$$

Hence ${\lambda }^{1/2}a$ has to be one of the zeroes of $j_{n,m}$ of $J_n$

Hence eigenvalues are given by

$$\lambda =\frac{j_{n,m}^2}{a^2},n=0,1,\cdots ,m=1,2,\cdots$$

with corresponding eigenfunctions

$$U_{n,m}(r)=J_n\left(j_{n,m}\left(\frac{r}{a}\right)\right)$$

Sturm-Liouville Form of Polar ODE

Let ODE be

$$\begin{array}{rl}\frac{1}{r}(r{U}_n^{\prime }(r){)}^{\prime }+\left(\lambda -\frac{n^2}{r^2}\right)U_n(r)& =0,\text{ for }0\le r<a\\ U_n(r)& =0,\text{ at }r=a\end{array}$$

from the motivation of Bessel’s Equation of Order $n$

Multiplying by $r$ and on disk of unit radius $a=1$ then

$$\begin{array}{rlrl}LU(r)=-(r{U}^{\prime }(r){)}^{\prime }+\frac{n^2}{r}U(r)& =\lambda rU(r),& & \text{ for }0\le r<1\\ U(r)& =0,& & \text{ at }r=1\\ U(r)& \text{ bounded }& & \text{ as }r\to 0\end{array}$$

Hence it has eigenvalues and eigenfunctions

$${\lambda }_{n,m}=j_{n,m}^2,U_{n,m}(r)=J_n(j_{n,m}r)$$

As it is a singular Sturm-Liouville equation with weighting function $r$ then

$${\int }_0^1{J}_n(j_{n,l}r)J_n(j_{n,m}r)rdr=0\text{ for }l\ne m$$

For $l=m$then

$${\int }_0^1{J}_n^2(j_{n,m}r)dr=\frac{1}{2}(J_n^{\prime }(j_{n,m}){)}^2$$