06. Special Functions

6.1 Introduction

Not Applicable

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6.2 Bessel Functions

6.2.1 Bessel’s Equation

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 $n$

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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

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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

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6.2.2 Bessel Functions of First and Second Kind

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

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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

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6.2.3 Properties of Bessel Functons

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 $Y_n$

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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 $n=0,1,2,3,4$

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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 $n=0,1,2,3,4$

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6.2.4 Normal Modes of a Circular Drum

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

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

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6.2.5 Sturm-Liouville Form

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

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

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6.3 Legendre Functions

6.3.1 The Legendre Equation

Motivation of Legendre Equation

Solving Helmholtz Equation using spherical polars $(r,\theta ,\varphi )$ so Laplacian is given by

$${\nabla }^{2}u=\frac{1}{r}\frac{{\partial }^2}{\partial r^2}(ru)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial u}{\partial \theta }\right)+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}u}{\partial {\varphi }^2}=-k^{2}u$$

Separating variables seek solution in form

$$u(r,\theta ,\varphi )=R(r)\Theta (\theta )\Phi (\varphi )$$

Rearranging equation to

$$\frac{r(rR(r){)}^{\prime \prime }}{R(r)}+k^2{r}^2=-\frac{(\sin \theta {\Theta }^{\prime }(\theta ){)}^{\prime }}{\sin \theta \Theta (\theta )}-\frac{{\Phi }^{\prime \prime }(\varphi )}{{\sin }^2\theta \Phi (\varphi )}$$

By usual argument then it must be equal to constant $\lambda$ so

$$-\frac{{\Phi }^{\prime \prime }(\varphi )}{\Phi (\varphi )}=\frac{\sin \theta (\sin \theta {\Theta }^{\prime }(\theta ){)}^{\prime }}{\Theta (\theta )}+\lambda {\sin }^2\theta$$

As $\Phi$ is $2\pi$-periodic function the constant is in form $m^2$ where $m\ge 0$ is an integer
Hence $\Phi =\text{const}$ if

$$m=0\text{ or }\Phi (\varphi )=\cos (m\varphi +\alpha )\text{ if }m>0$$

Hence results in following linear ODE for $\Theta (\theta )$

$$\frac{1}{\sin \theta }\frac{d}{d\theta }\left(\sin \theta \frac{d\Theta }{d\theta }\right)+\left(\lambda -\frac{m^2}{{\sin }^2\theta }\right)\Theta =0$$

As $\theta =0$ and $\theta =\pi$ are both regular singular points then

$$\Theta (\theta )\text{ is sufficiently well-behaved as }\theta \to 0,\pi$$

Hence using change of variables $\cos \theta =x$ and $\Theta (\theta )=y(x)$ with $\frac{d}{d\theta }=-\sin \theta \frac{d}{dx}$ so

$$\frac{d}{dx}\left((1-x^2)\frac{dy}{dx}\right)+\left(\lambda -\frac{m^2}{1-x^2}\right)y=0$$

which is known as the associated Legendre Equation for $y(x)$

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Associated Legendre Equation

$$\frac{d}{dx}\left((1-x^2)\frac{dy}{dx}\right)+\left(\lambda -\frac{m^2}{1-x^2}\right)y=0$$

where $m,\lambda$ can take any complex value generally

When $m=0$ then it is known as the Legendre Equation and Functions

Note that generally focusing on $m,l\in {\mathbb{Z}}^{>0}$ and $\lambda =l(l+1)$

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6.3.2 Properties of Legendre Functions

Properties of Legendre Functions

1. $x=\pm 1$ and $x=\infty$ are regular singular points of Associated Legendre Equation
   Indicial Exponents for $x=\pm 1$ are $-\frac{m}{2}$ and $\frac{m}{2}$
   Hence local expansion gives one bounded and unbounded solution as $x\to 1$ and $x\to -1$
   Note that if $m=0$, there is repeated root so solution is of order $\log (x\mp 1)$ as $x\to \pm 1$

2. Consider bounded solutions on $-1<x<1$ then
   Boundedness imposes two conditions, one at either end of interval
   Hence get posed singular Sturm-Liouville problem

$$\begin{array}{rlrl}-((1-x^2)y^{\prime }(x){)}^{\prime }+\frac{m^2}{1-x^2}y(x)& =\lambda y(x)& & \text{ for }-1<x<1\\ y(x)& \text{ bounded}& & \text{ as }x\to \pm 1\end{array}$$

3. Eigenvalues of Sturm-Liouville Form given by $\lambda =l(l+1)$ with integer $l\ge m\ge 0$
   Eigenfunctions are the corresponding associated Legendre functions denoted by

$$y(x)=P_l^m(x)$$

From Sturm-Liouville Theory then orthogonality relation is

$${\int }_{-1}^1{P}_k^m(x)P_l^m(x)dx=0\text{ for }k\ne l$$

4. For $m=0$ and integer $l\ge 0$ then
   Legendre functions are denoted by $P_l(x)$
   Hence $P_l$ is a polynomial of degree $l$ so if solution is as a power series expansion at $x=0$

$$y(x)=\sum _{k=0}^{\infty }{a}_k{x}^k$$

then series terminates with $a_k\equiv 0$ for $k>l$

Hence resulting Legendre Polynomials are given by Rodrigues’ Formula

$$P_l(x)=\frac{1}{2^{l}l!}\frac{d^l}{d{x}^l}[(x^2-1{)}^l]$$

5. Second, linearly independent solution of Legendre Equation $m=0$ is given by
   Legendre Function of Second Kind, denoted by $Q_l(x)$
   Solutions are unbounded as $x\to \pm 1$
   For case $l=0$ then solution $Q_0$ is

$$Q_0(x)=\frac{1}{2}\log \left(\frac{1+x}{1-x}\right)$$

6. General Case of nonzero $m\le l$ then
   Associated Legendre Functions of First and Second Kind are given by

$$\begin{array}{rl}{P}_l^m(x)& =(-1{)}^m(1-x^2{)}^{m/2}\frac{d^m{P}_l(x)}{d{x}^m}\\ Q_l^m(x)& =(-1{)}^m(1-x^2{)}^{m/2}\frac{d^m{Q}_l(x)}{d{x}^m}\end{array}$$

So associated Legendre Function $P_l^m$ is a polynomial if and only if $m$ is even

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6.4 Generalisation: Orthogonal Polynomials

Examples of Families of Orthogonal Polynomials as Solutions

Second Order Linear ODEs with Families of Orthogonal Polynomials as Solutions with

$${\int }_a^b{p}_m(x)p_n(x)r(x)d=0\text{ for }m\ne n$$

where fixed weighting function $r(x)$ can be inferred by appropriate Sturm-Liouville Eigenvalue Problem

Important Examples are

1. “Jacobi-like” polynomials in form (including Legendre, Chebyshev Polynomials)

$$(1-x^2)y^{\prime \prime }(x)+(a+bx)y^{\prime }(x)+\lambda y(x)=0$$

posed on interval $[-1,1]$ with constants $a,b$ and appropriate discrete set of values of $\lambda$

2. Associated Laguerre Polynomials satisfying Laguerre’s Equation

$$x{y}^{\prime \prime }(x)+(a+1-x)y^{\prime }(x)+\lambda y(x)=0$$

with polynomial solution $y(x)=L_n^a(x)$ where $\lambda \in {\mathbb{Z}}^{\ge 0}$
Satisfying Orthogonality Relation

$${\int }_0^{\infty }{L}_m^a(x)L_n^a(x)(x^a{e}^{-x})dx=0\text{ for }m\ne n$$

Laguerre Polynomials correspond to $a=0$ denoted by $L_n(x)\equiv L_n^0(x)$

3. Hermite Polynomials are solutions of Hermite Equation

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

which has polynomial solution $H_n(x)$ when $\lambda =2n$ for integer $n\ge 0$
Satisfying Orthogonality Relation

$${\int }_{-\infty }^{\infty }{H}_m(x)H_n(x)e^{-x^2}dx=0\text{ for }m\ne n$$

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